Recent zbMATH articles in MSC 47A https://zbmath.org/atom/cc/47A 2023-09-22T14:21:46.120933Z Werkzeug Common properties among various generalized inverses and constrained binary relations https://zbmath.org/1517.15003 2023-09-22T14:21:46.120933Z "Kuang, Ruifei" https://zbmath.org/authors/?q=ai:kuang.ruifei "Deng, Chunyuan" https://zbmath.org/authors/?q=ai:deng.chunyuan Summary: The common characterizations and various individual properties of different generalized inverses are established. Several equivalent conditions for the core-EP, weak group inverse, $$m$$-weak group inverse and weak core inverse are presented. The brand new explicit expressions for the operator binary relations defined by various generalized inverses are obtained. We derive properties and study the relationship between two different elements $$A, B \in \mathcal{B}(\mathcal{H})$$ and their $$(T, S)$$-outer generalized inverses $$A^{(2)}_{T, S}$$ and $$B^{(2)}_{T, S}$$ through which the reverse order law results about $$(AB)^{\bigcirc \!\!\!\!\!\!\mathrm{W}} = B^{\bigcirc \!\!\!\!\!\!\mathrm{W}} A^{\bigcirc \!\!\!\!\!\!\mathrm{W}}$$ and $$(AB)^{\bigcirc \!\!\!\!\!\mathrm{d}}=B^{\bigcirc \!\!\!\!\! \mathrm{d}} A^{\bigcirc \!\!\!\!\!\mathrm{d}}$$ are obtained. When does $$\Vert f(A) \vert = f(\Vert A \Vert)$$ hold true? https://zbmath.org/1517.15014 2023-09-22T14:21:46.120933Z "Bünger, Florian" https://zbmath.org/authors/?q=ai:bunger.florian "Rump, Siegfried M." https://zbmath.org/authors/?q=ai:rump.siegfried-michael The authors obtain conditions on norms on the set $$M_n$$ of complex $$n$$-by-$$n$$ matrices. They obtain the following result: Theorem. Let $$\|\cdot\|$$ be a norm on the set $$M_n$$, and let $$f(x):=\sum_{k=0}^{\infty}c_k x^k$$ with $$c_k>0$$, radius of convergence $$R\in (0,\infty]$$ and $$f(0)(\|I\|-1)=0$$. Suppose that one of the following cases holds true: the norm is \begin{itemize} \item[(1)] induced by a uniformly convex vector norm and $$c_k c_{k+1}\neq 0$$ for some $$k\geq 0$$; \item[(2)] unitarily invariant and $$c_k c_{k+1}\neq 0$$ for some $$k\geq 0$$; \item[(3)] the numerical radius and $$c_k c_{k+1}\neq 0$$ for some $$k\geq 1$$; \item[(4)] the $$\ell^{1}$$- or $$\ell^{\infty}$$-norm and $$c_k\neq 0$$ for all $$k\geq k_0$$ and some $$k_0\geq 0$$. \end{itemize} Then, for $$A\in M_n$$ with $$\|A\|\leq R$$, we have $$\|f(A)\|=f(\|A\|)$$ if and only if $$\|A\|$$ is an eigenvalue of $$A$$. This is a generalization of the result in [\textit{Yu. A. Abramovich} et al., J. Funct. Anal. 97, No. 1, 215--230 (1991; Zbl 0770.47005)] concerning the so-called Daugavet equation $$\|I+A\|=1+\|A\|$$. Reviewer: Takeaki Yamazaki (Kawagoe) An infinity norm bound for the inverse of strong $$\mathrm{SDD}_1$$ matrices with applications https://zbmath.org/1517.15015 2023-09-22T14:21:46.120933Z "Wang, Yinghua" https://zbmath.org/authors/?q=ai:wang.yinghua "Song, Xinnian" https://zbmath.org/authors/?q=ai:song.xinnian "Gao, Lei" https://zbmath.org/authors/?q=ai:gao.lei By imposing an additional condition on the main diagonal entries, the authors introduce a new class of matrices called strong SDD1 matrices (SDD stands for strictly diagonally dominant'', while SDD1 matrices were proposed by \textit{J. M. Peña} [Adv. Comput. Math. 35, No. 2--4, 357--373 (2011; Zbl 1254.65057)]) and present an infinity norm bound for the inverse of strong SDD1 matrices. Based on the proposed infinity norm bound, a new pseudospectra localization for matrices is obtained. Moreover, a new lower bound for the distance to instability is derived. Reviewer: Minghua Lin (Xi'an) Resonance frequencies of arbitrarily shaped dielectric cylinders https://zbmath.org/1517.35028 2023-09-22T14:21:46.120933Z "Shestopalov, Y." https://zbmath.org/authors/?q=ai:shestopaloff.yuri-k|shestopalov.yury-v|shestopalov.yuri-v (no abstract) Parametric generalization of the Meyer-König-Zeller operators https://zbmath.org/1517.41010 2023-09-22T14:21:46.120933Z "Sofyalıoğlu, Melek" https://zbmath.org/authors/?q=ai:sofyalioglu.melek "Kanat, Kadir" https://zbmath.org/authors/?q=ai:kanat.kadir "Çekim, Bayram" https://zbmath.org/authors/?q=ai:cekim.bayram Summary: The current paper deals with the parametric modification of Meyer-König-Zeller operators which preserve constant and Korovkin's other test functions in the form of $$(\frac{x}{1-x})^u$$, $$u=1,2$$ in limit case. The uniform convergence of the newly defined operators is investigated. The rate of convergence is studied by means of the modulus of continuity and by the help of Peetre-$$\mathcal{K}$$ functionals. Also, a Voronovskaya type asymptotic formula is given. Finally, some numerical examples are illustrated to show the effectiveness of the newly constructed operators for computing the approximation of function. A well-posed multidimensional rational covariance and generalized cepstral extension problem https://zbmath.org/1517.42005 2023-09-22T14:21:46.120933Z "Zhu, Bin" https://zbmath.org/authors/?q=ai:zhu.bin.6|zhu.bin|zhu.bin.1|zhu.bin.7|zhu.bin.4|zhu.bin.5 "Zorzi, Mattia" https://zbmath.org/authors/?q=ai:zorzi.mattia Summary: In the present paper we consider the problem of estimating the multidimensional power spectral density which describes a second-order stationary random field from a finite number of covariance and generalized cepstral coefficients. The latter can be framed as an optimization problem subject to multidimensional moment constraints, i.e., to search a spectral density maximizing an entropic index and matching the moments. In connection with systems and control, such a problem can also be posed as finding a multidimensional shaping filter (i.e., a linear time-invariant system) which can output a random field that has identical moments with the given data when fed with a white noise, a fundamental problem in system identification. In particular, we consider the case where the dimension of the random field is greater than two for which a satisfying theory is still missing. We propose a multidimensional moment problem which takes into account a generalized definition of the cepstral moments, together with a consistent definition of the entropy. We show that it is always possible to find a rational power spectral density matching exactly the covariances and approximately the generalized cepstral coefficients, from which a shaping filter can be constructed via spectral factorization. In plain words, our theory allows us to construct a well-posed spectral estimator for any finite dimension. On the continuous frame quantum detection problem https://zbmath.org/1517.42030 2023-09-22T14:21:46.120933Z "Hong, Guoqing" https://zbmath.org/authors/?q=ai:hong.guoqing "Li, Pengtong" https://zbmath.org/authors/?q=ai:li.pengtong Positive operator-valued measures (POVMs) play an important role in quantum detection where one has to recover a state from a collection of measurements from the system on this state. If a POVM can uniquely determine a state, it is called informationally complete. In the quantum detection problem, one seeks to characterize POVMs that are informationally complete. A quantum injective frame is a frame that can be used to distinguish states from their frame measurements, and the frame quantum detection problem seeks to characterize all such frames. Other authors have given some characterizations for the trace class and the Hilbert-Schmidt class injective continuous frames. In this paper, the authors present several characterizations for Schatten $$p$$-class injective continuous frames in terms of discrete representations of continuous frames, where $$1 \leq p < \infty$$. In that respect, the quantum detection problem is investigated in the general setting, and this work expands the scope of solutions to the quantum detection problem. Reviewer: Somantika Datta (Moscow, ID) A generalization of the Roberts orthogonality: from normed linear spaces to $$C^*$$-algebras https://zbmath.org/1517.46013 2023-09-22T14:21:46.120933Z "Faryad, Elias" https://zbmath.org/authors/?q=ai:faryad.elias "Khurana, Divya" https://zbmath.org/authors/?q=ai:khurana.divya "Moslehian, Mohammad Sal" https://zbmath.org/authors/?q=ai:moslehian.mohammad-sal "Sain, Debmalya" https://zbmath.org/authors/?q=ai:sain.debmalya The concept of orthogonality plays a pivotal role in efficiently describing the geometry of Banach spaces. Unlike the case of Hilbert spaces, there exist multiple definitions of orthogonality in a Banach space, which are not equivalent in general. Roberts orthogonality is historically the first type of orthogonality to be studied in the setting of Banach spaces, and it is useful in studying many geometric and analytic properties of the underlying space. In the present article, the authors introduce a generalization of the Roberts orthogonality, in the setting of $$C^*$$-algebras. In addition to studying the basic properties of the newly introduced concept, they establish an interesting connection between the Roberts orthogonality and the usual orthogonality in the case of certain special $$C^*$$-algebras, including the $$C^*$$-algebra of all $$2 \times 2$$ complex matrices. They also introduce a new concept of smoothness in normed linear spaces in terms of the additivity property of the usual Roberts orthogonality. A complete characterization of the same is obtained in the case of the classical $$\ell ^n_\infty$$ spaces. In view of the numerous recent works on the interconnection between orthogonality and the geometric properties of a Banach space, it is expected that the present article will be useful to specialists working on the topic of $$C^*$$-algebras. Reviewer: Kallol Paul (Kolkata) Stability of the inverses of interpolated operators with application to the Stokes system https://zbmath.org/1517.46018 2023-09-22T14:21:46.120933Z "Asekritova, I." https://zbmath.org/authors/?q=ai:asekritova.irina-u "Kruglyak, N." https://zbmath.org/authors/?q=ai:kruglyak.natan-ya "Mastyło, M." https://zbmath.org/authors/?q=ai:mastylo.mieczyslaw Summary: We study the stability of isomorphisms between interpolation scales of Banach spaces, including scales generated by well-known interpolation methods. We develop a general framework for compatibility theorems, and our methods apply to general cases. As a by-product we prove that the interpolated isomorphisms satisfy uniqueness-of-inverses. We use the obtained results to prove the stability of lattice isomorphisms on interpolation scales of Banach function lattices and demonstrate their application to the Calderón product spaces as well as to the real method scales. We also apply our results to prove solvability of the Neumann problem for the Stokes system of linear hydrostatics on an arbitrary bounded Lipschitz domain with a connected boundary in $$\mathbb{R}^n$$, $$n\geq 3$$, with data in some Lorentz spaces $$L^{p,q}(\partial \Omega, \mathbb{R}^n)$$ over the set $$\partial \Omega$$ equipped with a boundary surface measure. Free outer functions in complete Pick spaces https://zbmath.org/1517.46020 2023-09-22T14:21:46.120933Z "Aleman, Alexandru" https://zbmath.org/authors/?q=ai:aleman.alexandru "Hartz, Michael" https://zbmath.org/authors/?q=ai:hartz.michael "McCarthy, John E." https://zbmath.org/authors/?q=ai:mccarthy.john-e "Richter, Stefan" https://zbmath.org/authors/?q=ai:richter.stefan.1 Summary: \textit{M. T. Jury} and \textit{R. T. W. Martin} [Bull. Lond. Math. Soc. 51, No.~2, 223--229 (2019; Zbl 1435.46023); Indiana Univ. Math. J. 70, No.~1, 269--284 (2021; Zbl 1478.46022)] established an analogue of the classical inner-outer factorization of Hardy space functions. They showed that every function $$f$$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $$f=\varphi g$$, where $$g$$ is cyclic, $$\varphi$$ is a contractive multiplier, and $$\|f\|=\|g\|$$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization. Correction to: The $$(n+1)$$-centered operator on a Hilbert $$C^\ast$$-module'' https://zbmath.org/1517.46038 2023-09-22T14:21:46.120933Z "Liu, Na" https://zbmath.org/authors/?q=ai:liu.na "Xu, Qingxiang" https://zbmath.org/authors/?q=ai:xu.qingxiang "Zhang, Xiaofeng" https://zbmath.org/authors/?q=ai:zhang.xiaofeng From the text: The original version of this article [the authors, ibid. 17, No. 2, Paper No. 19, 17 p. (2023; Zbl 1514.46041)], published on 19 January 2023, unfortunately contained an error. Parenthesis should be present in citing all equations. The original article has been corrected. On the structure of the field $$C^\ast$$-algebra of a symplectic space and spectral analysis of the operators affiliated to it https://zbmath.org/1517.46051 2023-09-22T14:21:46.120933Z "Georgescu, Vladimir" https://zbmath.org/authors/?q=ai:georgescu.vladimir "Iftimovici, Andrei" https://zbmath.org/authors/?q=ai:iftimovici.andrei Summary: We show that the $$C^\ast$$-algebra generated by the field operators associated to a symplectic space $$\Xi$$ is graded by the semilattice of all finite dimensional subspaces of $$\Xi$$. If $$\Xi$$ is finite dimensional we give a simple intrinsic description of the components of the grading, we show that the self-adjoint operators affiliated to the algebra have a many channel structure similar to that of $$N$$-body Hamiltonians, in particular their essential spectrum is described by a kind of HVZ theorem, and we point out a large class of operators affiliated to the algebra. On a conjecture by Mbekhta about best approximation by polar factors https://zbmath.org/1517.47002 2023-09-22T14:21:46.120933Z "Chiumiento, Eduardo" https://zbmath.org/authors/?q=ai:chiumiento.eduardo Let $$\mathcal{B}(\mathcal{H})$$ be the algebra of all bounded linear operators on a complex separable Hilbert space $$\mathcal{H}$$, and $$\mathcal{I}$$ be the collection of all partial isometries on $$\mathcal{H}$$. The polar factor of an operator $$T\in \mathcal{B}(\mathcal{H})$$ is the unique $$V\in \mathcal{I}$$ such that $$T=V|T|$$ and $$\ker(V)=\ker(T)$$, where $$|T|:=(T^*T)^{1/2}$$ is the absolute value of $$T$$. In [J. Math. Anal. Appl. 487, No. 1, Article ID 123954, 12 p. (2020; Zbl 1486.47014)], \textit{M. Mbekhta} obtained explicit formulas the polar factor of any operator $$T\in\mathcal{B}(\mathcal{H})$$ and conjectured that, if $$X_0\in\mathcal{I}$$ such that $$\ker(X_0)=\ker(T)$$, then $$X_0$$ is the polar factor of $$T$$ if and only if $\|T-X_0\|=\min\{\|T-X\|:X\in\mathcal{I},\ \ker(X)=\ker(T)\}.$ In the aforecited paper, Mbekhta showed that the only if'' part holds true provided that $$T$$ is injective. \par Let $$P$$ and $$Q$$ be two orthogonal projections on $$\mathcal{H}$$ and set $j(P,Q):=\dim(\operatorname{ran}(P)\cap\ker(Q))-\dim(\ker(P)\cap\operatorname{ran}(Q))$ with the convention that $$j(P,Q)=0$$ if both dimensions are infinite. Here, $$\operatorname{ran}(T)$$ denotes the range of any operator $$T\in\mathcal{B}(\mathcal{H})$$. In the paper under review, the author shows that, if $$T\in\mathcal{B}(\mathcal{H})$$ with polar decomposition $$T=V|T|$$, then \begin{aligned} \|T-V\|&=\min\left\{\|T-X\|:X\in\mathcal{I},~j(V^*V,X^*X)\leq0 \right\}\\ &=\min\left\{\|T-X\|:X\in\mathcal{I},~j(VV^*,XX^*)\leq0 \right\}. \end{aligned} Since $$\ker(T)=\ker(V)$$ and $$j(V^*V,X^*X)=0$$ for any $$X\in\mathcal{I}$$ for which $$\ker(X)=\ker(T)$$, it then follows that $\|T-V\|=\min\{\|T-X\|:X\in\mathcal{I},~\ker(X)=\ker(T)\}.$ This shows that the only if'' part in Mbekhta's conjecture always holds without the injectivity condition on $$T$$. He also shows that the if'' part in such a conjecture is, in general, false. Moreover, he gives necessary and sufficient conditions on the spectrum of an operator to guarantee that its polar factor becomes a best approximate in the set of all partial isometries. Furthermore, he provides several examples and remarks which nicely illustrate the results obtained. Reviewer: Abdellatif Bourhim (Syracuse) A matrix formula for Schur complements of nonnegative selfadjoint linear relations https://zbmath.org/1517.47003 2023-09-22T14:21:46.120933Z "Contino, Maximiliano" https://zbmath.org/authors/?q=ai:contino.maximiliano "Maestripieri, Alejandra" https://zbmath.org/authors/?q=ai:maestripieri.alejandra-l "Marcantognini, Stefania" https://zbmath.org/authors/?q=ai:marcantognini.stefania-a-m Let $$A$$ be a nonnegative selfajoint linear relation in a Hilbert space $$\mathcal{H}$$ and $$\mathcal{S} \subset \mathcal{H}$$ be a closed subspace such that $$P_{\mathcal{S}} (\mathrm{dom}(A)) \subseteq \mathrm{dom} (A)$$, where $$P_{\mathcal{S}}$$ is the orthogonal projection of $$\mathcal{H}$$ onto $$\mathcal{S}$$. It is shown that $$A$$ admits a special matrix representation with respect to the decomposition $$\mathcal{S} \oplus \mathcal{S}^\bot$$. It is used to obtain formulas for the Schur complement of $$A$$ on $$\mathcal{S}$$ and for the $$\mathcal{S}$$-compression $$A_{\mathcal{S}}$$ of $$A$$. Reviewer: Valerii V. Obukhovskij (Voronezh) Operator factorization of range space relations https://zbmath.org/1517.47004 2023-09-22T14:21:46.120933Z "Joiţa, Maria" https://zbmath.org/authors/?q=ai:joita.maria "Costache, Tania-Luminiţa" https://zbmath.org/authors/?q=ai:costache.tania-luminita Summary: Given two range space relations $$A$$ and $$B$$ in Hilbert spaces, we characterize the existence of a range space operator $$T$$ such that $$A = BT$$, respectively $$A = TB$$. Spectra of upper triangular operator matrices with $$\mathrm{m}$$-complex symmetric operator entries along the main diagonal https://zbmath.org/1517.47005 2023-09-22T14:21:46.120933Z "Chrifi, S. Alaoui" https://zbmath.org/authors/?q=ai:chrifi.s-alaoui "Tajmouati, A." https://zbmath.org/authors/?q=ai:tajmouati.abdelaziz Summary: Let $$H$$ be a separable complex Hilbert space, a bounded linear operator $$T:H\to H$$ is $$\mathrm{m}$$-complex symmetric if there exists a conjugation $$J$$ on $$H$$ such that $$\Delta_m(T)=0$$ where $\Delta_m(T)=\sum_{k=0}^m (-1)^{m-k} \begin{pmatrix} m\\k \end{pmatrix} T^{\ast k}JT^{m-k}J$ In this paper, we study some spectral properties of m-complex symmetric operators like finite ascent and polaroid conditions. As a consequence, we examine Weyl-type theorems when the m-complex symmetric operator $$T$$ is polaroid. We also study the transition of spectral properties between two upper triangular operator matrices for which the diagonal entries are m-complex symmetric. Norm inequalities involving the weighted numerical radii of operators https://zbmath.org/1517.47006 2023-09-22T14:21:46.120933Z "Alrimawi, Fadi" https://zbmath.org/authors/?q=ai:alrimawi.fadi "Hirzallah, Omar" https://zbmath.org/authors/?q=ai:hirzallah.omar "Kittaneh, Fuad" https://zbmath.org/authors/?q=ai:kittaneh.fuad The authors give some norm inequalities involving the generalized weighted numerical radius $$w_{p,v}(\cdot)$$. They introduce norm inequalities that involve the generalized weighted numerical radius induced by the Schatten $$p$$-norm for certain type of $$2\times 2$$ block operator matrices. Reviewer: Elhadj Dahia (Bou Saâda) Anderson's theorem and $$A$$-spectral radius bounds for semi-Hilbertian space operators https://zbmath.org/1517.47007 2023-09-22T14:21:46.120933Z "Bhunia, Pintu" https://zbmath.org/authors/?q=ai:bhunia.pintu "Kittaneh, Fuad" https://zbmath.org/authors/?q=ai:kittaneh.fuad "Paul, Kallol" https://zbmath.org/authors/?q=ai:paul.kallol "Sen, Anirban" https://zbmath.org/authors/?q=ai:sen.anirban Summary: Let $$\mathcal{H}$$ be a complex Hilbert space and let $$A$$ be a positive bounded linear operator on $$\mathcal{H}$$. Let $$T$$ be an $$A$$-bounded operator on $$\mathcal{H}$$. For $$\mathrm{rank}(A) = n < \infty$$, we show that if $$W_A(T) \subseteq \overline{\mathbb{D}}( = \{\lambda \in \mathbb{C} : | \lambda | \leq 1 \})$$ and $$W_A(T)$$ intersects $$\partial \mathbb{D}( = \{\lambda \in \mathbb{C} : | \lambda | = 1 \})$$ at more than $$n$$ points, then $$W_A(T) = \overline{\mathbb{D}}$$. In particular, when $$A$$ is the identity operator on $$\mathbb{C}^n$$, then this leads to Anderson's theorem in the complex Hilbert space $$\mathbb{C}^n$$. We introduce the notion of $$A$$-compact operators to study analogous result when the space $$\mathcal{H}$$ is infinite dimensional. Further, we develop an upper bound for the $$\mathbb{A}$$-spectral radius of $$n \times n$$ operator matrices with entries are commuting $$A$$-bounded operators, where $$\mathbb{A} = \operatorname{diag}(A, A, \dots, A)$$ is an $$n \times n$$ diagonal operator matrix. Several inequalities involving $$A$$-spectral radius of $$A$$-bounded operators are also given. Convexity of the orbit-closed $$C$$-numerical range and majorization https://zbmath.org/1517.47008 2023-09-22T14:21:46.120933Z "Loreaux, Jireh" https://zbmath.org/authors/?q=ai:loreaux.jireh "Patnaik, Sasmita" https://zbmath.org/authors/?q=ai:patnaik.sasmita Summary: We introduce and investigate the orbit-closed $$C$$-numerical range, a natural modification of the $$C$$-numerical range of an operator introduced for $$C$$ trace-class by \textit{G. Dirr} and \textit{F. vom Ende} [Linear Multilinear Algebra 68, No. 4, 652--678 (2020; Zbl 1502.47010)]. Our orbit-closed $$C$$-numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when $$C$$ is finite rank. Since Dirr and vom Ende's results concerning the $$C$$-numerical range depend only on its closure, our orbit-closed $$C$$-numerical range inherits these properties, but we also establish more. For $$C$$ self-adjoint, Dirr and vom Ende were only able to prove that the closure of their $$C$$-numerical range is convex and asked whether it is convex without taking the closure. We establish the convexity of the orbit-closed $$C$$-numerical range for self-adjoint $$C$$ without taking the closure by providing a characterization in terms of majorization, unlocking the door to a plethora of results which generalize properties of the $$C$$-numerical range known in finite dimensions or when $$C$$ has finite rank. Under rather special hypotheses on the operators, we also show the $$C$$-numerical range is convex, thereby providing a partial answer to the question posed by Dirr and vom Ende. Numerical radius and Berezin number inequality https://zbmath.org/1517.47009 2023-09-22T14:21:46.120933Z "Majee, Satyabrata" https://zbmath.org/authors/?q=ai:majee.satyabrata "Maji, Amit" https://zbmath.org/authors/?q=ai:maji.amit "Manna, Atanu" https://zbmath.org/authors/?q=ai:manna.atanu The authors study various inequalities for the numerical radius and the Berezin number of a bounded linear operator on a Hilbert space. The numerical radius of a pure two-isometry is 1 and the Crawford number of a pure two-isometry is 0. For any scalar-valued non-constant inner function $$\theta$$, the numerical radius and the Crawford number of a Toeplitz operator $$T_\theta$$ on a Hardy space is $$1$$ and $$0$$, respectively. It is also shown that the numerical radius is multiplicative for a class of isometries and sub-multiplicative for a class of commutants of a shift. These results are illustrated with some concrete examples. At the end, some Hardy-type inequalities for Berezin number of certain class of operators are established with the help of the classical Hardy inequality. Reviewer: V. Lokesha (Bangalore) $$M_\varphi$$-type submodules over the bidisk https://zbmath.org/1517.47010 2023-09-22T14:21:46.120933Z "Yang, Guo Zeng" https://zbmath.org/authors/?q=ai:yang.guozeng "Wu, Chang Hui" https://zbmath.org/authors/?q=ai:wu.changhui "Yu, Tao" https://zbmath.org/authors/?q=ai:yu.tao Summary: Let $$H^2(\mathbb{D}^2)$$ be the Hardy space over the bidisk $$\mathbb{D}^2$$, and let $$M_\varphi=[(z-\varphi(w))^2]$$ be the submodule generated by $$(z-\varphi(w))^2$$, where $$\varphi(w)$$ is a function in $$H^\infty(w)$$. The related quotient module is denoted by $$N_\varphi=H^2(\mathbb{D}^2)\ominus M_\varphi$$. In the present paper, we study the Fredholmness of compression operators $$S_z,S_w$$ on $$N_\varphi$$. When $$\varphi(w)$$ is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator $$F_w$$ on $$[(z-w)^2]\ominus z[(z-w)^2]$$ and the Beurling type theorem holds for the fringe operator $$F_z$$ on $$M_\varphi\ominus wM_\varphi$$ if $$\varphi(0)=0$$. Lastly, we study some properties of $$F_w$$ on $$[(z-w^2)^2]\ominus z[(z-w^2)^2]$$. Supercyclicity of weighted composition operators on spaces of continuous functions https://zbmath.org/1517.47011 2023-09-22T14:21:46.120933Z "Beltrán-Meneu, M. J." https://zbmath.org/authors/?q=ai:beltran-meneu.maria-j "Jordá, E." https://zbmath.org/authors/?q=ai:jorda.enrique "Murillo-Arcila, M." https://zbmath.org/authors/?q=ai:murillo-arcila.marina|arcila.m-murillo Let $$\mathcal{C}(X)$$ denote the space of continuous functions on a topological Hausdorff space $$X$$, and denote by $$\tau _{p}$$ the topology of pointwise convergence on $$X$$. Let $$E$$ be a locally convex space continuously embedded in $$(\mathcal{C}(X),\tau _{p})$$, and such that the functionals on $$E$$ of point evaluation $$\delta _{x}$$, $$x\in X$$, are linearly independent in the dual space $$E'$$ of $$E$$. The authors study the dynamics of weighted composition operators $$C_{w,\,\varphi }$$ on $$E$$. Here, $$C_{w,\varphi }:f\mapsto w\,.\,f\circ \varphi$$, where $$w:X\to \mathbb{C}$$ is the multiplier, and $$\varphi :X\to X$$ is a continuous function called the symbol. The main dynamical property under study is that of weak supercyclicity: the operator $$C_{w,\varphi }$$ is weakly supercyclic if there exists a function $$f\in E$$ such that the set $\{\lambda \,C_{w,\,\varphi }^{n}:\lambda \in\mathbb{C},\ n\ge 0\}$ is weakly dense in $$E$$. The authors prove that, if $$X$$ is compact, $$E$$ is a Banach space containing a nowhere vanishing function, and if $$E$$ embeds continuously in $$(\mathcal{C}(X),\|\cdot\|_{\infty})$$, then $$C_{w,\varphi }$$ is never weakly supercyclic on $$E$$. They also obtain that a weighted composition operator is never $$\tau _{p}$$-supercyclic on the disk algebra $$A(\mathbb{D})$$, nor on the analytic Lipschitz spaces $$\mathrm{Lip}_{\alpha }(\mathbb{D})$$, $$0<\alpha \le 1$$, and that there are no weakly supercyclic composition operators on the space of holomorphic functions on $$\mathbb{C}\setminus\{0\}$$ and $$\mathbb{D}\setminus\{0\}$$, respectively. Reviewer: Sophie Grivaux (Villeneuve d'Ascq) Chaos for convolution operators on the space of entire functions of infinitely many complex variables https://zbmath.org/1517.47012 2023-09-22T14:21:46.120933Z "Caraballo, Blas M." https://zbmath.org/authors/?q=ai:caraballo.blas-melendez "Favaro, Vinicius V." https://zbmath.org/authors/?q=ai:favaro.vinicius-vieira Summary: In sharp contrast to a classical result of \textit{G. Godefroy} and \textit{J. H. Shapiro} [J. Funct. Anal. 98, No. 2, 229--269 (1991; Zbl 0732.47016)], Mujica and the second author showed that no translation operator on the space $$\mathcal{H}(\mathbb{C}^\mathbb{N})$$ of entire functions of infinitely many complex variables is hypercyclic [\textit{V. V. Fávaro} and \textit{J. Mujica}, J. Oper. Theory 76, No. 1, 141--158 (2016; Zbl 1399.47038)]. In an attempt to better understand the dynamics of such operators, in this work we show, firstly, that no convolution operator on $$\mathcal{H}(\mathbb{C}^\mathbb{N})$$ is cyclic or $$n$$-supercyclic for any positive integer $$n$$. In the opposite direction, we show that every non-trivial convolution operator on $$\mathcal{H}(\mathbb{C}^\mathbb{N})$$ is mixing. Particularizing Arai's concept of Li-Yorke chaos to non-metrizable topological vector spaces, we show that non-trivial convolution operators on $$\mathcal{H}(\mathbb{C}^\mathbb{N})$$ are also Li-Yorke chaotic. Hypercyclicity and supercyclicity of unbounded operators https://zbmath.org/1517.47013 2023-09-22T14:21:46.120933Z "Kumar, Abhay" https://zbmath.org/authors/?q=ai:kumar.abhay Summary: We study hypercyclicity and supercyclicity of unbounded operators with special focus on generators of composition $$C_0$$-semigroups and give conditions under which they are supercyclic and non-supercyclic. Further, we show that if $$A$$ is a closed range operator with $$0 \in \sigma (A)$$, then the sufficient conditions of the continuous version of Godefroy and Shapiro's Criterion, which is given in [\textit{S. E. Mourchid}, Semigroup Forum 73, No. 2, 313--316 (2006; Zbl 1115.47009)] for hypercyclicity, are necessary as well. When is a scaled contraction hypercyclic? https://zbmath.org/1517.47014 2023-09-22T14:21:46.120933Z "Matache, Valentin" https://zbmath.org/authors/?q=ai:matache.valentin Let $$H$$ be a separable, infinite-dimensional complex Hilbert space and let $$T$$ be a bounded linear operator on $$H$$. The author studies the set $$\Lambda(T)$$ (which I would like to call the hypercyclicity range of $$T$$) of all scalars $$\lambda\in \mathbb C$$ for which $$S:=\lambda T$$ is a hypercyclic operator (meaning that there is $$x\in H$$ for which its orbit $$\{S^nx: n\in \mathbb N\}$$ is dense in $$H$$). Badea, Grivaux and Mueller, e.g., unveiled an invertible bilateral weighted shift $$T$$ on $$\ell_2(\mathbb Z)$$ such that $$T$$ and $$3T$$ are hypercyclic, but not $$2T$$ [\textit{C. Badea} et al., Proc. Am. Math. Soc. 137, No. 4, 1397--1403 (2009; Zbl 1168.47007)]. Many known results on the hypercyclicity of these scalar multiples are revisited, making the whole a nice, easily readable survey. It is also shown that for any hypercyclic operator $$T$$ on $$H$$, every connected component of the essential spectrum $$\sigma_e(T)$$ meets the unit circle, extending the corresponding result for the spectrum itself. In particular, if $$T$$ is essentially quasi-nilpotent, then $$\Lambda(T)=\Lambda(T^*)=\emptyset$$. Reviewer: Raymond Mortini (Metz) Dilations of commuting $$C_0$$-semigroups with bounded generators and the von Neumann polynomial inequality https://zbmath.org/1517.47015 2023-09-22T14:21:46.120933Z "Dahya, Raj" https://zbmath.org/authors/?q=ai:dahya.raj Summary: Consider $$d$$ commuting $$C_0$$-semigroups (or equivalently: $$d$$-parameter $$C_0$$-semigroups) over a Hilbert space for $$d \in \mathbb{N}$$. In the literature [\textit{B. Sz.-Nagy} and \textit{C. Foiaş}, Harmonic analysis of operators on Hilbert space. Budapest: Akadémiai Kiadó; Amsterdam-London: North-Holland Publishing Company (1970; Zbl 0201.45003); \textit{M. Slocinski}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 22, 1011--1014 (1974; Zbl 0302.47032); Zesz. Nauk. Uniw. Jagielloń. 623, Pr. Mat. 23, 191--194 (1982; Zbl 0515.47019); \textit{M. Ptak}, Ann. Pol. Math. 45, 237--243 (1985; Zbl 0606.47040); \textit{C. Le Merdy}, Indiana Univ. Math. J. 45, No. 4, 945--959 (1996; Zbl 0876.47009); \textit{E. Shamovich} and \textit{V. Vinnikov}, Integral Equations Oper. Theory 87, No. 1, 45--80 (2017; Zbl 06715517)], conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focusing on semigroups with bounded generators, we establish a simple and natural condition on the generators, viz.\ complete dissipativity, which naturally extends the basic notion of the dissipativity of the generators. Using examples of non-doubly commuting semigroups, this property can be shown to be strictly stronger than dissipativity. As the first main result, we demonstrate that complete dissipativity completely characterises the existence of regular unitary dilations, and extend this to the case of arbitrarily many commuting $$C_0$$-semigroups. We furthermore show that all multi-parameter $$C_0$$-semigroups (with bounded generators) admit a weaker notion of regular unitary dilations, and provide simple sufficient norm criteria for complete dissipativity. The paper concludes with an application to the von Neumann polynomial inequality problem, which we formulate for the semigroup setting and solve negatively for all $$d \geqslant 2$$. Operator theory on noncommutative polydomains. II https://zbmath.org/1517.47016 2023-09-22T14:21:46.120933Z "Popescu, Gelu" https://zbmath.org/authors/?q=ai:popescu.gelu This paper continues the study of noncommutative polydomains and their universal operator models generated by admissible $$k$$-tuples of formal power series in several noncommuting indeterminates. As was announced in Part~I [\textit{G. Popescu}, Complex Anal. Oper. Theory 16, No. 4, Paper No. 50, 101 p. (2022; Zbl 1516.47024)], the results obtained there will be used here to develop a dilation theory for non-pure elements in admissible noncommutative polydomains and to obtain a complete description for the invariant subspaces of the corresponding universal operator models. The paper starts with recalling some necessary notions and notations made in the first part, and introducing other ones. So, for the $$k$$-tuple $$\mathbf{g}=(\mathbf{g}_1,\dots,\mathbf{g}_k)$$ of free holomorphic functions in a neighborhood of the origin in the operator ball $$B(\mathcal{H})^{n_i}$$, the Hilbert space $$F^2(\mathbf{g})$$ of formal power series is attached. $$\mathcal{M}_{\mathbf{g}}(\mathcal{H})$$ is the noncommutative set of all $$k$$-tuples $$X=(X_1,\dots,X_k)$$ in $$B(\mathcal{H})^{n_1}\times\cdots\times B(\mathcal{H})^{n_k}$$, with $$X_i = (X_{i,1}, \dots, X_{i,n_i})$$. The defect operator $$\Delta_{\mathbf{g}^{-1}}(X,X^*)$$ is defined and a pure noncommutative polydomain $$\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})$$ is a subset of $$B(\mathcal{H})^{n_1+\dots+n_k}$$ as the set of pure solutions of operator inequation given by the defect operator $$\Delta_{\mathbf{g}^{-1}}(X,X^*)\ge0$$. The universal model $$\mathbf{W}$$ for $$X$$ is defined and the pure part and the Cuntz part $$\mathcal{D}^{c}_{\mathbf{g}^{-1}}(\mathcal{H})$$ in the noncommutative polydomain $$\mathcal{D}_{\mathbf{g}^{-1}}(\mathcal{H})$$ are introduced. The role of universal operator model associated with $$\mathbf{g}$$ will be played by a $$k$$-tuple $$\mathbf{W} = (\mathbf{W}_1,\dots,\mathbf{W}_k)$$ with $$\mathbf{W}_i := (\mathbf{W}_{i,1},\dots,\mathbf{W}_{i,n_i})$$, where $$\mathbf{W}_{i,j}$$ are weighted left creation operators acting on the tensor products $$F^2(H_{n_1})\otimes \dots\otimes F^2(H_{n_k})$$, where $$F^2(H_{n_i})$$ is the full Fock space with $$n_i$$ generators. A condition for $$\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})=\mathcal{M}_{\mathbf{g}}(\mathcal{H})$$ was found, and in this case $$\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})$$ is called an admissible polydomain and $$\mathbf{g}=(\mathbf{g}_1,\dots,\mathbf{g}_k)$$ is an admissible $$k$$-tuple for operator model theory. Also, some examples of classes of $$k$$-tuples of formal power series admissible for operator model theory are recalled. The second section of the paper is devoted to $$C^*$$-algebras associated with noncommutative polydomains and Wold decompositions. It is shown that there is a unique minimal nontrivial two-sided ideal of the $$C^*$$-algebra $$C^*(\mathbf{W})$$ generated by the operators $$\mathbf{W}_{i,j}$$ and the identity, namely, the ideal $$\mathcal{K}$$ of all compact operators in $$B(\bigotimes_{s=1}^k F^2(H_{n_s}))$$. A geometric version of the Wold decomposition for unital $$*$$-representations of the $$C^*$$-algebra $$C^*(W)$$ is obtained. This extends the corresponding result previous obtained by the author in this more general setting. In the third section, the notions of completely non-pure $$k$$-tuple $$X$$ from $$\mathcal{D}_{\mathbf{g}^{-1}}(\mathcal{H})$$, pure and completely non-pure representation $$\pi$$, and Cuntz-type representation are defined. The Cuntz-type algebra $$\mathcal{O}(\mathbf{g})$$ is introduced as the universal $$C^*$$-algebra generated by $$\pi(\mathbf{W}_{i,s})$$ and the identity, where $$\pi$$ is a completely non-pure $$*$$-representation of $$C^*(\mathbf{W})$$. Using the Wold decompositions from the previous section, an exact sequence of $$C^*$$-algebras generalizing the well-known results obtained by \textit{L. A. Coburn} for the unilateral shift [Bull. Am. Math. Soc. 73, 722--726 (1967; Zbl 0153.16603)] and by \textit{J. Cuntz} for the left creation operators [Commun. Math. Phys. 57, 173--185 (1977; Zbl 0399.46045)] on the full Fock space with $$n$$ generators is obtained. In the fourth section, it is shown that, for any admissible noncommutative polydomain $$\mathcal{D}_{\mathbf{g}^{-1}}$$, the corresponding universal model $$\mathbf{W} = (\mathbf{W}_1,\dots,\mathbf{W}_k)$$ with $$\mathbf{W}_i := (\mathbf{W}_{i,1},\dots,\mathbf{W}_{i,n_i})$$ admits a Beurling-type characterization of the joint invariant subspaces of $$\mathbf{W}_{i,j}$$. In the fifth section, a~functional model for the pure elements in admissible noncommutative polydomains is provided. To each $$X$$ in $$\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})$$, a~characteristic function $$\Theta_{\mathbf{g},X} := (\Theta^{(1)}_{\mathbf{g},X},\dots, \Theta^{(k)}_{\mathbf{g},X})$$ is associated, which consists of partially isometric multi-analytic operators, and it is shown that $$X$$ is unitarily equivalent to $$G = (G_1,\dots, G_n)$$ with $$G_i := (G_{i,1},\dots,G_{i,n_i})$$ of an appropriate form. It is also proved that, if $$T\in \mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})$$ and $$T'\in \mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H}')$$, then $$T$$ and $$T'$$ are unitarily equivalent if and only if their characteristic functions $$\Theta_{\mathbf{g},T}$$ and $$\Theta_{\mathbf{g},T'}$$ coincide in a certain sense. The dilation theory on admissible noncommutative polydomains is analyzed in the last section of the paper. For an admissible tuple $$\mathbf{g}$$, it is proved that, if $$X\in \overline{\mathcal{D}^{\mathrm{pure}}_{\mathbf{g}^{-1}}(\mathcal{H})}$$, then there is a $$*$$-representation $$\pi : C^*(\mathbf{W}) \rightarrow\mathcal{K}_\pi$$ on a separable Hilbert space $$\mathcal{K}_\pi$$ such that $$\pi$$ annihilates the compact operators in $$C^*(\mathbf{W})$$ and $$\Delta_{\mathbf{g}^{-1}}(\pi(\mathbf{W}), \pi(\mathbf{W})^*) = 0$$, and $$\mathcal{H}$$ can be identified with a $$*$$-cyclic co-invariant subspace of $$\mathcal{K} := (\bigotimes^k_{s=1}F^2(H_{n_s} ) \otimes\mathcal{D})\oplus\mathcal{K}_\pi$$ under the operators $$(\mathbf{W}_{i,j}\otimes I_{\mathcal{D}})\oplus\pi(\mathbf{W}_{i,j})$$ such that $$X^*_{i,j}=[(\mathbf{W}_{i,j}\otimes I_{\mathcal{D}})\oplus\pi(\mathbf{W}_{i,j})]_{\mathcal{H}}$$ for $$i\in \{1,\dots,k\}$$, $$j \in \{1,\dots,n_i\}$$, where $$\mathcal{D} := \overline{\Delta_{\mathbf{g}^{-1}} (X, X^*)\mathcal{H}}$$. The uniqueness of the dilation is also analyzed. Reviewer: Ilie Valuşescu (Bucureşti) Belonging of Gel'fand integral of positive operator valued functions to separable ideals of compact operators on Hilbert space https://zbmath.org/1517.47017 2023-09-22T14:21:46.120933Z "Krstić, Mihailo" https://zbmath.org/authors/?q=ai:krstic.mihailo-a "Milović, Matija" https://zbmath.org/authors/?q=ai:milovic.matija "Milošević, Stefan" https://zbmath.org/authors/?q=ai:milosevic.stefan The authors study the weak$$^\ast$$ integral of operator-valued function. They find sufficient and necessary conditions for these operators to belong to ideals of compact linear operators. Some applications of the results are given. Reviewer: Elhadj Dahia (Bou Saâda) On Berezin number inequalities for operator matrices https://zbmath.org/1517.47018 2023-09-22T14:21:46.120933Z "Sahoo, Satyajit" https://zbmath.org/authors/?q=ai:sahoo.satyajit "Das, Namita" https://zbmath.org/authors/?q=ai:das.namita "Rout, Nirmal Chandra" https://zbmath.org/authors/?q=ai:rout.nirmal-chandra For a bounded linear operator, acting in the reproducing kernel Hilbert space $$\mathcal{H}=\mathcal{H}(\Omega)$$ over some set $$\Omega$$, its Berezin symbol $$\tilde{A}$$ is defined by $$\tilde{A}(\lambda)=\langle A\tilde{k}_\lambda, \tilde{k}_\lambda \rangle$$, where $$\tilde{k}_\lambda$$ the normalized reproducing kernel of $$\mathcal{H}$$. The Berezin set and the Berezin number of an operator $$A$$ are defined, respectively, by $\text{Ber}(A):=\{\tilde{A}(\lambda): \lambda\in\Omega \}\quad \hbox{and} \quad \text{ber}(A):= \sup \{\tilde{A}(\lambda): \lambda\in\Omega \}.$ Clearly, $$\mathrm{Ber}(A)$$ is contained in the numerical range of $$A$$ and so $$\mathrm{ber}(A)$$ is no larger than the numerical radius of $$A$$. Results related to the numerical range and numerical radius could be similarly considered for the Berezin set and the Berezin number. This paper presents refinements of certain earlier existing bounds for the Berezin number of operator matrices and proves some new Berezin number inequalities for general $$n\times n$$ operator matrices. Further, it also establishes several upper bounds for the Berezin number and the generalized Euclidean Berezin number (introduced by \textit{M. Bakherad} [Czech. Math. J. 68, No. 4, 997--1009 (2018; Zbl 1482.47003)] for off-diagonal operator matrices. Reviewer: Minghua Lin (Xi'an) $$(m, C)$$-isometric Toeplitz operators with rational symbols https://zbmath.org/1517.47019 2023-09-22T14:21:46.120933Z "Ko, Eungil" https://zbmath.org/authors/?q=ai:ko.eungil "Lee, Ji Eun" https://zbmath.org/authors/?q=ai:lee.jieun.1 "Lee, Jongrak" https://zbmath.org/authors/?q=ai:lee.jun-ik Summary: An operator $$T\in \mathcal{L}(\mathcal{H})$$ is said to be $$(m, C)$$-isometric if there exists a conjugation $$C$$ such that $\sum\limits^m_{j=0} (-1)^{m-j} \begin{pmatrix} m \\ j \end{pmatrix} T^{*j} CT^J C=0$ for some positive integer $$m$$. In this paper, we study $$(m, C)$$-isometric Toeplitz operators $$T_{\varphi}$$ with rational symbols $$\varphi$$. We characterize $$(m, C)$$-isometric Toeplitz operators $$T_{\varphi}$$ by properties of the rational symbols $$\varphi$$. Moreover, we provide a concrete description of the $$(m,\mathcal{C})$$-isometric block Toeplitz operators. Factorizations of contractions https://zbmath.org/1517.47020 2023-09-22T14:21:46.120933Z "Das, B. Krishna" https://zbmath.org/authors/?q=ai:krishna-das.b "Sarkar, Jaydeb" https://zbmath.org/authors/?q=ai:sarkar.jaydeb "Sarkar, Srijan" https://zbmath.org/authors/?q=ai:sarkar.srijan Summary: The celebrated Sz.-Nagy and Foiaş theorem asserts that every pure contraction is unitarily equivalent to an operator of the form $$P_QM_z|_Q$$ where $$Q$$ is a $$M_z^\ast$$-invariant subspace of a $$\mathcal{D}$$-valued Hardy space $$H_{\mathcal{D}}^2(\mathbb{D})$$, for some Hilbert space $$\mathcal{D}$$. On the other hand, the celebrated theorem of Berger, Coburn and Lebow [\textit{C. A. Berger} et al., J. Funct. Anal. 27, 51--99 (1978; Zbl 0383.46010)] on pairs of commuting isometries can be formulated as follows: a~pure isometry $$V$$ on a Hilbert space $$\mathcal{H}$$ is a product of two commuting isometries $$V_1$$ and $$V_2$$ in $$\mathcal{B}(\mathcal{H})$$ if and only if there exist a Hilbert space $$\mathcal{E}$$, a unitary $$U$$ in $$\mathcal{B}(\mathcal{E})$$ and an orthogonal projection $$P$$ in $$\mathcal{B}(\mathcal{E})$$ such that $$(V,V_1,V_2)$$ and $$(M_z,M_\Phi,M_{\Psi})$$ on $$H_{\mathcal{E}}^2(\mathbb{D})$$ are unitarily equivalent, where $\Phi(z)=(P+zP^\bot)U^\ast\text{ and }\Psi(z)=U(P^\bot zP)\; (z\in \mathbb{D}).$ In this context, it is natural to ask whether similar factorization results hold true for pure contractions. The purpose of this paper is to answer this question. More specifically, let $$T$$ be a pure contraction on a Hilbert space $$\mathcal{H}$$ and let $$P_QM_z|_Q$$ be the Sz.-Nagy and Foiaş representation of $$T$$ for some canonical $$\mathcal{Q} \subseteq H_D^2(\mathbb{D})$$. Then $$T=T_1T_2$$, for some commuting contractions $$T_1$$ and $$T_2$$ on $$\mathcal{H}$$, if and only if there exist $$\mathcal{B}(\mathcal{D})$$-valued polynomials {$$\phi$$} and {$$\psi$$} of degree such that $$\mathcal{Q}$$ is a joint $$(M_\phi^\ast,M_\psi^\ast$$-invariant subspace, $P_QM_z|_Q=P_QM_{{\phi}{\psi}}|_Q=P_QM_{{\psi}{\phi}}|_Q$ and $(T_1,T_2)\cong P_QM_{\phi}|_Q,P_QM_{\psi}|_Q).$ Moreover, there exist a Hilbert space $$E$$ and an isometry $$V \in B(D;E)$$ such that $\phi (z)=V^\ast \Phi(z)V\text{ and }\psi(z)=V^\ast\Psi(z)V \; (z\in \mathbb{D}),$ where the pair $$(\Phi,\Psi)$$, as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on $$H_{\mathcal{E}}^2(\mathbb{D})$$. As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions. Wold-type decomposition for $$\mathcal{U}_n$$-twisted contractions https://zbmath.org/1517.47021 2023-09-22T14:21:46.120933Z "Majee, Satyabrata" https://zbmath.org/authors/?q=ai:majee.satyabrata "Maji, Amit" https://zbmath.org/authors/?q=ai:maji.amit Summary: Let $$n > 1$$, and $$\{U_{i j} \}$$ for $$1 \leq i < j \leq n$$ be $$\binom{n}{2}$$ commuting unitaries on a Hilbert space $$\mathcal{H}$$ such that $$U_{j i} : = U_{i j}^\ast$$. An $$n$$-tuple of contractions $$(T_1, \ldots, T_n)$$ on $$\mathcal{H}$$ is called $$\mathcal{U}_n$$-twisted contraction with respect to a twist $$\{U_{i j} \}_{i < j}$$ if $$T_1, \ldots, T_n$$ satisfy $T_i T_j = U_{i j} T_j T_i, \ T_i^\ast T_j = U_{i j}^\ast T_j T_i^\ast \ \text{and}\ T_k U_{i j} = U_{i j} T_k$ for all $$i,j, k = 1, \ldots, n$$ and $$i \neq j$$. We obtain a recipe to calculate the orthogonal spaces of the Wold-type decomposition for $$\mathcal{U}_n$$-twisted contractions on Hilbert spaces. As a by-product, a new proof as well as complete structure for $$\mathcal{U}_2$$-twisted (or pair of doubly twisted) and $$\mathcal{U}_n$$-twisted isometries have been established. Left-invertibility of rank-one perturbations https://zbmath.org/1517.47022 2023-09-22T14:21:46.120933Z "Das, Susmita" https://zbmath.org/authors/?q=ai:das.susmita "Sarkar, Jaydeb" https://zbmath.org/authors/?q=ai:sarkar.jaydeb Summary: For each isometry $$V$$ acting on some Hilbert space and a pair of vectors $$f$$ and $$g$$ in the same Hilbert space, we associate a nonnegative number $$c(V; f, g)$$ defined by $c(V; f, g) = (\|f\|^2 - \|V^\ast f\|^2)\|g\|^2 + |1 + \langle V^\ast f, g\rangle|^2.$ We prove that the rank-one perturbation $$V + f \otimes g$$ is left-invertible if and only if $c(V;f,g) \neq 0.$ We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function $$z$$. Finally, we examine $$D + f \otimes g$$, where $$D$$ is a diagonal operator with nonzero diagonal entries and $$f$$ and $$g$$ are vectors with nonzero Fourier coefficients. We prove that $$D + f\otimes g$$ is left-invertible if and only if $$D+f\otimes g$$ is invertible. Necessary and sufficient conditions for $$n$$-times Fréchet differentiability on $$\mathcal{S}^p$$, $$1<p<\infty$$ https://zbmath.org/1517.47023 2023-09-22T14:21:46.120933Z "Le Merdy, Christian" https://zbmath.org/authors/?q=ai:le-merdy.christian "McDonald, Edward" https://zbmath.org/authors/?q=ai:mcdonald.edward-a For $$1<p<\infty$$, let $$\mathcal{S}^p(H)$$ the $$p$$-Schatten class on the infinite-dimensional separable Hilbert space $$H$$. A Lipschitz continuous function $$f:\mathbb{R} \to\mathbb{C}$$ is said to be \begin{itemize} \item $$1$$-time Fréchet differentiable on $$\mathcal{S}^p(H)$$ at $$A$$ (self-adjoint operator on $$H$$) if there exists a bounded linear map $$D_p^1 f(A) \in \mathcal{B}(\mathcal{S}^p, \mathcal{S}^p)$$ such that $$\|f(A+X) -f(A)-D_p^1 f(A)[X]\|_p =o(\|X\|_p)$$ as $$\|X\|_p \rightarrow 0$$, $$X \in \mathcal{S}^p_{sa}$$; \item $$n$$-times Fréchet differentiable on $$\mathcal{S}^p(H)$$ at $$A$$ if $$f$$ is $$(n-1)$$-times Fréchet differentiable on $$\mathcal{S}^p(H)$$ at $$A+X$$, for every $$X$$ in a $$\mathcal{S}^p_{sa}$$-neighborhood of $$0$$ and there exists a bounded $$n$$-multilinear operator $$D_p^n f(A)\in \mathcal{B}_n(\mathcal{S}^p \times \ldots \times\mathcal{S}^p, \mathcal{S}^p)$$ such that \begin{align*} &\|D_p^{n-1} f(A+X)[X_1, \dots, X_{n-1}]- D_p^{n-1} f(A)[X_1, \dots, X_{n-1}] - D_p^{n} f(A)[X_1, \dots, X_{n-1}, X]\|_p =\\ & 0(\|X_1\|_p, \dots, \|X_{n-1}\|_p, \|X\|_p) \end{align*} as $$\|X\|_p \rightarrow 0$$, $$X \in \mathcal{S}^p_{sa}$$, uniformly for all $$X_1, \dots X_{n-1} \in \mathcal{S}^p$$; \item $$n$$-times continuously Fréchet differentiable on $$\mathcal{S}^p(H)$$ if it is $$n$$-times Fréchet differentiable on $$\mathcal{S}^p(H)$$ at every self-adjoint operator $$A$$ and the map $X \mapsto D_p^n f(A+X)$ is continuous. \end{itemize} The main result of the paper states that, for a Lipschitz continuous function $$f$$ on $$\mathbb{R}$$, the following are equivalent: \begin{itemize} \item[(i)] $$f$$ is $$n$$-times continuously Fréchet differentiable on $$\mathcal{S}^p(H)$$; \item[(ii)] for any self-adjoint $$A$$, $$f$$ is $$n$$-times Fréchet differentiable at $$A$$; \item[(iii)] $$f$$ is $$n$$-times Fréchet differentiable, $$f^\prime, f^{\prime \prime}, \dots, f^{(n)}$$ are bounded, and $$f^{(n)}$$ is uniformly continuous. \end{itemize} Reviewer: Daniele Puglisi (Catania) Integral operators for nonlocally compact group modules https://zbmath.org/1517.47024 2023-09-22T14:21:46.120933Z "Ludkowski, Sergey Victor" https://zbmath.org/authors/?q=ai:ludkovsky.sergey-victor Summary: The article is devoted to integral operators for nonlocally compact group modules. Operator valued functions for nonlocally compact groups and their normed spaces of different types are investigated. For this purpose quasi-invariant measures on nonlocally compact groups are used. Estimates of norms of integral operators are given. Moreover, convolutions of functions having operator values and values in Banach spaces are scrutinized. Examples are given. Functional calculi for sectorial operators and related function theory https://zbmath.org/1517.47025 2023-09-22T14:21:46.120933Z "Batty, Charles" https://zbmath.org/authors/?q=ai:batty.charles-j-k "Gomilko, Alexander" https://zbmath.org/authors/?q=ai:gomilko.alexandre-m "Tomilov, Yuri" https://zbmath.org/authors/?q=ai:tomilov.yuri Summary: We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them. Two approaches to the use of unbounded operators in Feynman's operational calculus https://zbmath.org/1517.47026 2023-09-22T14:21:46.120933Z "Nielsen, Lance" https://zbmath.org/authors/?q=ai:nielsen.lance \textit{A. E. Taylor} [Acta Math. 84, 189--224 (1951; Zbl 0042.12103)] introduced a functional calculus for unbounded operators using methods from complex analysis, whereas Kato's functional calculus [\textit{T. Kato}, Perturbation theory for linear operators. Reprint of the corr. print. of the 2nd ed. 1980. Berlin: Springer-Verlag (1995; Zbl 0836.47009)] involves the use of Kato's analytic families of closed unbounded operators. In this article, the author investigates these two approaches to the use of unbounded operators in Feynman's operational calculus on a Banach algebra. The author applies Feynman's operational calculus using the Taylor calculus approach to discuss the connection between the operational calculus and the modified Feynman integral of \textit{M. Lapidus} [J. Funct. Anal. 63, 261--275 (1985; Zbl 0601.47025)] and \textit{G. W. Johnson} and \textit{M. L. Lapidus} [The Feynman integral and Feynman's operational calculus. Oxford: Clarendon Press (2000; Zbl 0952.46044)]. Also discussed is the stability of the operational calculus for unbounded operators obtained using Taylor's and the analytic family approach. Reviewer: Santhosh Kumar Pamula (Bangalore) Note on the assumptions in working with generalized inverses https://zbmath.org/1517.47027 2023-09-22T14:21:46.120933Z "Cvetković Ilić, Dragana S." https://zbmath.org/authors/?q=ai:cvetkovic-ilic.dragana-s Summary: Most of the published results on solving operator equations are very restrictive, i.e., they have been proved under certain additional assumptions and in fact we do not have general solvability conditions. There are many reasons why this is so, one of them being the fact that the usual methods employed when solving these equations involve the use of generalized inverses which exist and are bounded in the case of operators only under certain special conditions such as closedness of the range of operators. Using two previously considered systems of operator equations as examples, we will show that using a particular general approach we can give necessary and sufficient solvability conditions without any additional assumptions. We will consider the system $$A X C = C = C X A$$ and generalize recent particular results from [\textit{C.-Y. Deng} et al., Appl. Math. Lett. 81, 86--92 (2018; Zbl 1503.15013)], as well as $$B A X = B = X A B$$ for which the particular results are given in [\textit{C. Deng}, J. Math. Anal. Appl. 398, No. 2, 664--670 (2013; Zbl 1303.47002)]. We intend to use our approach to initiate formulating various general solvability conditions for other systems of operator equations. On Rubel's problem in the class of linear operators on the space of analytic functions https://zbmath.org/1517.47028 2023-09-22T14:21:46.120933Z "Linchuk, Yu. S." https://zbmath.org/authors/?q=ai:linchuk.yu-s Summary: In this paper, we describe all derivation pairs of linear operators that act in spaces of functions analytic in domains. Further inequalities involving the weighted geometric operator mean and the Heinz operator mean https://zbmath.org/1517.47029 2023-09-22T14:21:46.120933Z "Al-Subaihi, Ibrahim Ahmed" https://zbmath.org/authors/?q=ai:al-subaihi.ibrahim-ahmed "Raïssouli, Mustapha" https://zbmath.org/authors/?q=ai:raissouli.mustapha Summary: In this paper, we first investigate some inequalities involving the $$p$$-weighted geometric operator mean $A\sharp_p B=A^{1/2} \Big( A^{-1/2}BA^{-1/2}\Big)^p A^{1/2},$ where $$p \in [0, 1]$$ is a real number and $$A, B$$ are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to $$A\sharp_p B$$ with a high rate of convergence. Some relationships involving $$A\sharp_p B$$ are deduced. Numerical examples illustrating the theoretical results are also discussed. Operator Jensen's type inequalities for convex functions https://zbmath.org/1517.47030 2023-09-22T14:21:46.120933Z "Hosseini, Mohsen Shah" https://zbmath.org/authors/?q=ai:hosseini.mohsen-shah "Moradi, Hamid Reza" https://zbmath.org/authors/?q=ai:moradi.hamid-reza "Moosavi, Baharak" https://zbmath.org/authors/?q=ai:moosavi.baharak Summary: This paper is mainly devoted to studying operator Jensen inequality. More precisely, a new generalization of Jensen inequality and its reverse version for convex (not necessary operator convex) functions have been proved. Several special cases are discussed as well. Reversing Bellman operator inequality https://zbmath.org/1517.47031 2023-09-22T14:21:46.120933Z "Sababheh, Mohammad" https://zbmath.org/authors/?q=ai:sababheh.mohammad-s "Moradi, Hamid Reza" https://zbmath.org/authors/?q=ai:moradi.hamid-reza "Furuichi, Shigeru" https://zbmath.org/authors/?q=ai:furuichi.shigeru The Bellman inequality for Hilbert space operators has been presented in [\textit{A. Morassaei} et al., Linear Algebra Appl. 438, No. 10, 3776--3780 (2013; Zbl 1301.47026)] and [\textit{M. Bakherad} and \textit{A. Morassaei}, Indag. Math., New Ser. 26, No. 4, 646--659 (2015; Zbl 1334.47020)]. The authors of the present paper use the Mond-Pečarić method to prove that, for a given scalar $$\alpha>0$$, there exists a scalar $$\beta$$ such that the inequality $$\alpha(\Phi(I-A\nabla_\nu B))^{1/p}+\beta I \leq\Phi ((I-A)^{1/p}\nabla_\nu(I-B)^{1/p})$$ holds for any positive invertible operators $$A, B\in\mathbb{B}(\mathscr{H})$$, a~normalized positive linear map $$\Phi$$ on $$\mathbb{B}(\mathscr{H})$$, $$1\leq\nu\leq 1$$, and $$p>1$$, and thus obtain some multiplicative and additive reverses of the operator Bellman inequality. Reviewer: Mohammad Sal Moslehian (Mashhad) Quadratic weighted geometric mean in Hermitian unital Banach $$\ast$$-algebras https://zbmath.org/1517.47032 2023-09-22T14:21:46.120933Z "Dragomir, S. S." https://zbmath.org/authors/?q=ai:dragomir.sever-silvestru Summary: In this paper we introduce the quadratic weighted geometric mean $x\circledS_{\nu}y := \Big||yx^{-1}|^vx\Big|^2$ for invertible elements $$x, y$$ in a Hermitian unital Banach $$\ast$$-algebra and real number $$\nu$$. We show that $x\circledS_{\nu}y = |x|^2\sharp_{\nu}|y|^2,$ where $$\sharp_{\nu}$$ is the usual geometric mean and provide some inequalities for this mean under various assumptions for the elements involved. Factorization of ordinary and hyperbolic integro-differential equations with integral boundary conditions in a Banach space https://zbmath.org/1517.47033 2023-09-22T14:21:46.120933Z "Providas, Efthinios" https://zbmath.org/authors/?q=ai:providas.efthinios "Pulkina, Ludmila Stepanovna" https://zbmath.org/authors/?q=ai:pulkina.ludmila-stepanovna "Parasidis, Ioannis Nestorios" https://zbmath.org/authors/?q=ai:parasidis.ioannis-nestorios Summary: The solvability condition and the unique exact solution by the universal factorization (decomposition) method for a class of abstract operator equations of the type $B_1u=\mathcal{A}u-S\Phi(A_0u)-GF(\mathcal{A}u)=f ,\quad u\in D(B_1),$ where $$\mathcal{A}, A_0$$ are linear abstract operators, $$G, S$$ are linear vectors and $$\Phi, F$$ are linear functional vectors is investigated. This class is useful for solving boundary value problems (BVPs) with integro-differential equations (IDEs), where $$\mathcal{A}, A_0$$ are differential operators and $$F(\mathcal{A}u), \Phi(A_0u)$$ are Fredholm integrals. It is shown that the operators of the type $$B_1$$ can be factorized in the some cases in the product of two more simple operators $$B_G, B_{G_0}$$ of special form, which are derived analytically. Further, the solvability condition and the unique exact solution for $$B_1u=f$$ easily follow from the solvability condition and the unique exact solutions for the equations $$B_G v=f$$ and $$B_{G_0}u=v$$. Operators which preserve a positive definite inner product https://zbmath.org/1517.47034 2023-09-22T14:21:46.120933Z "Andruchow, Esteban" https://zbmath.org/authors/?q=ai:andruchow.esteban Summary: Let $$\mathcal{H}$$ be a Hilbert space, $$A$$ a positive definite operator in $$\mathcal{H}$$ and $$\langle f,g\rangle_A=\langle Af,g\rangle$$, $$f, g\in \mathcal{H}$$, the $$A$$-inner product. This paper studies the geometry of the set $\mathcal{I}_A^a:=\{\text{adjointable isometries for } \langle\ ,\ \rangle_A\}.$ It is proved that $$\mathcal{I}_A^a$$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in $$\mathcal{H}$$ which are unitaries for the $$A$$-inner product. Smooth curves in $$\mathcal{I}_A^a$$ with given initial conditions, which are minimal for the metric induced by $$\langle\ ,\ \rangle_A$$, are presented. This result depends on an adaptation of M.~G. Krein's method for the lifting of symmetric contractions in order that it works also for symmetrizable transformations (i.e., operators which are selfadjoint for the $$A$$-inner product). Fuglede-Putnam theorem and quasisimilarity of class $$p-wA(s,t)$$ operators https://zbmath.org/1517.47035 2023-09-22T14:21:46.120933Z "Chō, M." https://zbmath.org/authors/?q=ai:cho.myunghyun|cho.minho|cho.minsung|cho.minhaeng|cho.manwon|cho.minjae|cho.minkyoung|cho.moonki|cho.minshik|cho.muneo|cho.myeonghwan|cho.manki|cho.minjeong|cho.maoquan|cho.maenghyo|cho.misung|cho.moonhee|cho.minsik|cho.meehyung|cho.mann|cho.miyoung "Prasad, T." https://zbmath.org/authors/?q=ai:prasad.t-ram|prasad.t-v-s-r-k|prasad.tribhuan|prasad.t-b-aruna|prasad.t-jayachandra "Rashid, M. H. M." https://zbmath.org/authors/?q=ai:rashid.mohammad-hussein-mohammad|rashid.malik-h-m "Tanahashi, K." https://zbmath.org/authors/?q=ai:tanahashi.kotoro|tanahashi.katsumi|tanahashi.kotaro "Uchiyama, A." https://zbmath.org/authors/?q=ai:uchiyama.atsushi|uchiyama.atsuko|uchiyama.akihito|uchiyama.akihiko Summary: We show that $$p-wA(s,t)$$ operators $$S,T^\ast$$ ($$s + t \leqslant 1$$, $$0 < p \leqslant 1$$), with $$\ker(S) \subseteq \ker(S^\ast)$$ and $$\ker(T^\ast) \subseteq \ker(T)$$ satisfy the Fuglede-Putnam theorem, i.e., $$SX = XT$$ for some $$X$$ implies $$S^\ast X= XT^\ast$$. Also, we show that two quasisimilar $$p-wA(s,t)$$ operators $$S,T$$ ($$s+t \leqslant 1$$, $$0 < p \leqslant 1$$) with $$\ker(S) \subseteq \ker(S^\ast)$$ and $$\ker(T) \subseteq \ker(T^\ast)$$ have equal spectra and essential spectra. Certain properties involving the unbounded operators $$p(T)$$, $$TT^\ast$$, and $$T^\ast T$$; and some applications to powers and \textit{nth} roots of unbounded operators https://zbmath.org/1517.47036 2023-09-22T14:21:46.120933Z "Mortad, Mohammed Hichem" https://zbmath.org/authors/?q=ai:mortad.mohammed-hichem Summary: In this paper, we are concerned with conditions under which $$[ p (T) ]^\ast = \overline{p}(T^\ast)$$, where $$p(z)$$ is a one-variable complex polynomial, and $$T$$ is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the identities $$\sigma(A B) = \sigma(B A)$$, where $$A$$ and $$B$$ are two unbounded operators. The equations $$(T T^\ast)^\ast = T T^\ast$$ and $$(T^\ast T)^\ast = T^\ast T$$, where $$T$$ is a densely defined closable operator, are also studied. A particular interest will be paid to the equation $$T^\ast T = p(T)$$ and its variants. Then, we have certain results concerning \textit{nth} roots of classes of normal and nonnormal (unbounded) operators. Some further consequences and counterexamples accompany our results. Spectra of weighted composition operators with quadratic symbols https://zbmath.org/1517.47038 2023-09-22T14:21:46.120933Z "Doctor, Jessica" https://zbmath.org/authors/?q=ai:doctor.jessica "Hodges, Timothy" https://zbmath.org/authors/?q=ai:hodges.timothy-j "Kaschner, Scott" https://zbmath.org/authors/?q=ai:kaschner.scott-r "McFarland, Alexander" https://zbmath.org/authors/?q=ai:mcfarland.alexander "Thompson, Derek" https://zbmath.org/authors/?q=ai:thompson.derek-a Summary: Previously, spectra of certain weighted composition operators $$W_{\psi, \varphi}$$ on $$H^2$$ were determined under one of two hypotheses: either $$\varphi$$ converges under iteration to the Denjoy-Wolff point uniformly on all of $$\mathbb{D}$$ rather than simply on compact subsets, or $$\varphi$$ is essentially linear fractional.'' We show that if $$\varphi$$ is a quadratic self-map of $$\mathbb{D}$$ of parabolic type, then the spectrum of $$W_{\psi, \varphi}$$ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so. Norms of hyponormal weighted composition operators on the Hardy and weighted Bergman spaces https://zbmath.org/1517.47039 2023-09-22T14:21:46.120933Z "Fatehi, Mahsa" https://zbmath.org/authors/?q=ai:fatehi.mahsa "Shaabani, Mahmood Haji" https://zbmath.org/authors/?q=ai:shaabani.mahmood-haji Summary: In this paper, first we find norms of hyponormal weighted composition operators $$C_{\psi,\varphi}$$, when $$\varphi$$ has a Denjoy-Wolff point on the unit circle. Then for $$\varphi$$ which is an analytic selfmap of $$\mathbb D$$ with a fixed point in $$\mathbb D$$, we investigate norms of hyponormal weighted composition operators $$C_{\psi,\varphi}$$. Conjectures on spectra of composition operators and related issues https://zbmath.org/1517.47040 2023-09-22T14:21:46.120933Z "Gallardo-Gutiérrez, Eva A." https://zbmath.org/authors/?q=ai:gallardo-gutierrez.eva-a "Matache, Valentin" https://zbmath.org/authors/?q=ai:matache.valentin Summary: A conjecture posed by \textit{C. C. Cowen} and \textit{B. D. MacCluer} in [Composition operators on spaces of analytic functions. Boca Raton, FL: CRC Press (1995; Zbl 0873.47017)] states that the spectrum of composition operators acting on the Hardy space $$H^2$$ induced by analytic selfmaps of the open unit disc having a fixed point in that disc, other than the identity or elliptic automorphisms, is always representable as the union of a connected set containing the origin, the so-called Schröder eigenvalues of the inducing selfmap, and 1. Our first result is proving that the aforementioned conjecture holds. We also consider two related conjectures regarding the spectra of composition operators, proving that one of them holds, and showing that the other one (which is still open) is satisfied in particular cases. Weighted integral Hankel operators with continuous spectrum https://zbmath.org/1517.47046 2023-09-22T14:21:46.120933Z "Fedele, Emilio" https://zbmath.org/authors/?q=ai:fedele.emilio "Pushnitski, Alexander" https://zbmath.org/authors/?q=ai:pushnitski.alexander-b Summary: Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in $$L^2(\mathbb{R}_+)$$. These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel $$s^\alpha t^\alpha (s + t)^{-1-2 \alpha}$$, where $$\alpha > -1/2$$. Our analysis can be considered as an extension of \textit{J. S. Howland}'s 1992 paper [Indiana Univ. Math. J. 41, No. 2, 427--434 (1992; Zbl 0773.47013)] which dealt with the unweighted case, corresponding to $$\alpha = 0$$. Minimal reducing subspaces of $$k^{th}$$ order slant Toeplitz operators https://zbmath.org/1517.47047 2023-09-22T14:21:46.120933Z "Hazarika, Munmun" https://zbmath.org/authors/?q=ai:hazarika.munmun "Marik, Sougata" https://zbmath.org/authors/?q=ai:marik.sougata Summary: In this paper we have identified the reducing and minimal reducing subspaces of the $$k^{th}$$ order slant Toeplitz operator induced by $$\varphi(z) = z^N$$, where $$N$$ is an integer. Self-commutator norm of hyponormal Toeplitz operators https://zbmath.org/1517.47049 2023-09-22T14:21:46.120933Z "Le, Trieu" https://zbmath.org/authors/?q=ai:le.trieu Summary: \textit{C. Chu} and \textit{D. Khavinson} [Proc. Am. Math. Soc. 144, No. 6, 2533--2537 (2016; Zbl 1350.30075)] obtained a lower bound for the norm of the self-commutator of a certain class of hyponormal Toeplitz operators on the Hardy space. Via a different approach, we offer a generalization of their result. On unitary equivalence to a self-adjoint or doubly-positive Hankel operator https://zbmath.org/1517.47051 2023-09-22T14:21:46.120933Z "Martin, Robert T. W." https://zbmath.org/authors/?q=ai:martin.robert-t-w Summary: Let $$A$$ be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry $$V$$ so that $$AV > 0$$ and $$A$$ is Hankel with respect to $$V$$, i.e., $$V^*A = AV$$, if and only if $$A$$ is not invertible. The isometry $$V$$ can be chosen to be isomorphic to $$N \in \mathbb{N} \cup\{+\infty\}$$ copies of the unilateral shift if $$A$$ has spectral multiplicity at most $$N$$. We further show that the set of all isometries $$V$$ so that $$A$$ is Hankel with respect to $$V$$ are in bijection with the set of all closed, symmetric restrictions of $$A^{-1}$$. Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion https://zbmath.org/1517.47053 2023-09-22T14:21:46.120933Z "Wang, Yi" https://zbmath.org/authors/?q=ai:wang.yi.33 "Xia, Jingbo" https://zbmath.org/authors/?q=ai:xia.jingbo Suppose that $$\mathcal{N}$$ is a either a submodule or a quotient module of a Hilbert module $$\mathcal{H}$$. Let $$P_{\mathcal{N}}:\mathcal{H}\to\mathcal{N}$$ be the orthogonal projection and define $$\mathcal{Z}_{\mathcal{N},j}= P_{\mathcal{N}}M_{z_j}|\mathcal{N}$$ for $$j=1,\dots,n$$. Recall that $$\mathcal{N}$$ is said to be $$p$$-essentially normal if all commutators $$[\mathcal{Z}_{\mathcal{N},i}^*,\mathcal{Z}_{\mathcal{N},j}]$$ with $$1\le i,j\le n$$ are in the Schatten class $$\mathcal{C}_p$$. Let $$\mathbb{B}=\{z\in\mathbb{C}^n:|z|<1\}$$ and $$\mathbb{S}=\{z\in\mathbb{C}^n:|z|=1\}$$, where $$\mathbb{S}$$ is equipped with the standard spherical measure $$d\sigma$$. The Hardy space $$H^2(\mathbb{S})$$ is the closure of the ring of analytic polynomials $$\mathbb{C}[z_1,\dots,z_n]$$ in $$L^2(\mathbb{S},d\sigma)$$. Let $$\Omega$$ be a complex manifold. A~subset $$A\subset\Omega$$ is called a complex analytic subset of $$\Omega$$ if, for each $$a\in A$$, there are a neighborhood $$U$$ of $$a$$ and functions $$f_1,\dots,f_N$$ analytic in this neighborhood such that $$A\cap U=\{z\in U:f_1(z)=\ldots=f_N(z)=0\}$$. Let $$\widetilde{M}$$ be an analytic subset of an open neighborhood of $$\overline{\mathbb{B}}$$ with $$1\le\dim_{\mathbb{C}}\widetilde{M}\le n-1$$ and $$M=\overline{B}\cap\widetilde{M}$$. Consider a submodule $$\mathcal{R}=\{f\in H^2(\mathbb{S}):f=0 \text{ on } M\}$$ and the corresponding quotient module $$\mathcal{Q}=H^2(\mathbb{S})\ominus\mathcal{R}$$. The authors prove the geometric Arveson-Douglas conjecture saying that the quotient module $$\mathcal{Q}$$ is $$p$$-essentially normal for every $$p>d=\dim_{\mathbb{C}}\widetilde{M}$$. Further, for a measure $$\mu$$ on $$M$$, one can define the Toeplitz operator $$T_\mu$$ by $$(T_\mu h)(z)=\int_M h(w)(1-\langle z,w\rangle)^{-n}\,d\mu(w)$$. Let $$Q$$ denote the orthogonal projection from $$L^2(\mathbb{S},d\sigma)$$ onto $$\mathcal{Q}$$. The second main result says that, for constants $$0<c\le C<\infty$$, there exists a measure $$\mu$$ on $$M$$ such that $$cQ\le T_\mu\le CQ$$ on $$L^2(\mathbb{S},d\sigma)$$. For each $$f\in L^\infty(\mathbb{S},d\sigma)$$, define $$Q_f=QM_f|\widetilde{Q}$$. Let $$\mathcal{TQ}$$ be the $$C^*$$-algebra generated by $$\{Q_f:f\in L^\infty(\mathbb{S},d\sigma)\}$$. The third main result of the paper says that, if $$A\in\mathcal{TQ}$$ and $$\lim_{z\in M,|z|\to 1}\langle Ak_z,k-z\rangle=0$$, where $$k_z$$ is the reproducing kernel for $$H^2(\mathbb{S})$$, then $$A$$ is a compact operator. Reviewer: Oleksiy Karlovych (Lisboa) Perturbation determinants and discrete spectra of semi-infinite non-self-adjoint Jacobi operators https://zbmath.org/1517.47055 2023-09-22T14:21:46.120933Z "Golinskii, Leonid B." https://zbmath.org/authors/?q=ai:golinskii.leonid-b Summary: We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb-Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed. Periodic perturbations of unbounded Jacobi matrices. III: The soft edge regime https://zbmath.org/1517.47056 2023-09-22T14:21:46.120933Z "Świderski, Grzegorz" https://zbmath.org/authors/?q=ai:swiderski.grzegorz Summary: We present a pretty detailed spectral analysis of Jacobi matrices with periodically modulated entries in the case when 0 lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that the studied operators are always self-adjoint irrespective of the modulated sequence. Moreover, if the growth of the modulated sequence is superlinear, then the spectrum of the considered operators is always discrete. Finally, we study regular perturbations of this class in the linear and the sublinear cases. We impose conditions assuring that the spectrum is absolutely continuous on some regions of the real line. A~constructive formula for the density in terms of Turán determinants is also provided. For Parts I see [the author and \textit{B. Trojan}, J. Approx. Theory 216, 38--66 (2017; Zbl 1367.47040)], for Part II [the author, ibid 216, 67--85 (2017; Zbl 1367.47039)]. Backward extensions of weighted shifts on directed trees https://zbmath.org/1517.47058 2023-09-22T14:21:46.120933Z "Pikul, Piotr" https://zbmath.org/authors/?q=ai:pikul.piotr Summary: The weighted shifts are long known and form an important class of operators. One of generalisations of this class are weighted shifts on directed trees, where the linear order of coordinates in $$\ell^2$$ is replaced by a more involved graph structure. In this paper, we focus on the question of joint backward extending of a given family of weighted shifts on directed trees to a weighted shift on an enveloping directed tree that preserves subnormality or power hyponormality of considered operators. One of the main results shows that the existence of such a joint backward extension'' for a family of weighted shifts on directed trees depends only on the possibility of backward extending of single weighted shifts that are members of the family. We introduce a generalised framework of weighted shifts on directed forests (disjoint families of directed trees) which seems to be more convenient to work with. A~characterisation of leafless directed forests on which all hyponormal weighted shifts are power hyponormal is given. The spectrum and fine spectrum of generalized Rhaly-Cesàro matrices on $$c_0\ \mathrm{and}\ c$$ https://zbmath.org/1517.47059 2023-09-22T14:21:46.120933Z "Yildirim, Mustafa" https://zbmath.org/authors/?q=ai:yildirim.mustafa "Mursaleen, Mohammad" https://zbmath.org/authors/?q=ai:mursaleen.mohammad "Doğan, Çağla" https://zbmath.org/authors/?q=ai:dogan.cagla Summary: The generalized Rhaly Cesàro matrices $$A_{\alpha}$$ are the triangular matrix with nonzero entries $$a_{nk} = \alpha^{n-k}/(n + 1)$$ with $$\alpha \in [0,1]$$. In [Proc. Am. Math. Soc. 86, 405--409 (1982; Zbl 0505.47021)], \textit{H. C. Rhaly jun.}\ determined boundedness, compactness of generalized Rhaly Cesàro matrices on $$\ell_2$$ Hilbert space and showed that its spectrum is $$\sigma(A_{\alpha},\ell_2) = \{1/n\} \cup \{0\}$$. Also, in [Linear Multilinear Algebra 26, No. 1--2, 49--58 (1990; Zbl 0697.15009)], lower bounds for these classes were obtained under certain restrictions on $$\ell_p$$ by \textit{B. E. Rhoades}. In the present paper, boundedness, compactness, spectra, the fine spectra and subdivisions of the spectra of generalized Rhaly Cesàro operator on $$c_0\ \mathrm{and}\ c$$ have been determined. Nonlinear maps preserving condition spectrum of Jordan skew triple product of operators https://zbmath.org/1517.47068 2023-09-22T14:21:46.120933Z "Benbouziane, H." https://zbmath.org/authors/?q=ai:benbouziane.hassane "Bouramdane, Y." https://zbmath.org/authors/?q=ai:bouramdane.y "El Kettani, M. Ech-Cherif" https://zbmath.org/authors/?q=ai:el-kettani.m-ech-cherif "Lahssaini, A." https://zbmath.org/authors/?q=ai:lahssaini.aziz Summary: Let $$\mathcal{B(H)}$$ the algebra of all bounded linear operators on a complex Hilbert space $$\mathcal H$$ with $$\dim \mathcal H \geqslant 3$$. Let $$\mathcal{W,V}$$ be subsets of $$\mathcal{B(H)}$$ which contain all rank-one operators. Denote by $$r_{\varepsilon}(A)$$ the condition spectral radius of $$A \in \mathcal{B(H)}$$. We determine the form of surjective maps $$\phi : \mathcal W \rightarrow \mathcal V$$ satisfying $$r_{\varepsilon}(AB^*A) = r_{\varepsilon}(\phi(A)\phi(B)^*\phi(A))$$ for all $$A,B$$ in $$\mathcal W$$, we characterize also the structure of surjective maps $$\phi : \mathcal{B(H)} \rightarrow \mathcal{B(H)}\ \mathrm{with}\ \sigma_{\varepsilon}(AB^*A) = \sigma_{\varepsilon}(\phi(A)\phi(B)^*\phi(A))$$ for all $$A,B$$ in $$\mathcal{B(H)}$$ where $$\sigma_{\varepsilon}(A)$$ is the $$\varepsilon$$-condition spectrum of an operator $$A$$ in $$\mathcal{B(H)}$$. The $$L^p$$ boundedness of the wave operators for matrix Schrödinger equations https://zbmath.org/1517.47075 2023-09-22T14:21:46.120933Z "Weder, Ricardo" https://zbmath.org/authors/?q=ai:weder.ricardo-a Summary: We prove that the wave operators for $$n \times n$$ matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $$L^p (\mathbb{R}^+, \mathbb{C}^n)$$, $$1 < p < \infty$$, for slowly decaying selfadjoint matrix potentials $$V$$ that satisfy the condition $$\int_0^{{\infty}} (1 + x) |V (x)| \,d x < {\infty}$$. Moreover, assuming that $$\int_0^{{\infty}} (1 + x^\gamma) |V (x)|\,d x < {\infty}$$, $$\gamma > \frac{5}{2}$$, and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $$L^1 (\mathbb{R}^+, \mathbb{C}^n)$$ and in $$L^{{\infty}} (\mathbb{R}^+, \mathbb{C}^n)$$. We also prove that the wave operators for $$n \times n$$ matrix Schrödinger equations on the line are bounded in the spaces $$L^p (\mathbb{R}, \mathbb{C}^n)$$, $$1 < p < {\infty}$$, assuming that the perturbation consists of a point interaction at the origin and of a potential $$\mathcal{V}$$ that satisfies the condition $$\int_{-\infty}^{\infty} (1+|x|)|\mathcal{V}(x)|\, dx<\infty$$. Further, assuming that $$\int_{-\infty}^{\infty} (1+|x|^\gamma)|\mathcal{V}(x)| \,dx<\infty$$, $$\gamma > \frac{5}{2}$$, and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $$L^1 (\mathbb{R}, \mathbb{C}^n)$$ and in $$L^{{\infty}} (\mathbb{R}, \mathbb{C}^n)$$. We obtain our results for $$n \times n$$ matrix Schrödinger equations on the line from the results for $$2 n \times 2 n$$ matrix Schrödinger equations on the half line. Norm-ideal perturbations of one-parameter semigroups and applications https://zbmath.org/1517.47077 2023-09-22T14:21:46.120933Z "Boulton, Lyonell" https://zbmath.org/authors/?q=ai:boulton.lyonell "Dimoudis, Spyridon" https://zbmath.org/authors/?q=ai:dimoudis.spyridon Firstly, the authors examine a general framework for perturbation of generators relative to the Schatten-von Neumann ideals on Hilbert spaces, obtaining in this way a development for a family of equivalence relations on generators. Also, they prove applications of the abstract setting to heat semigroups associated with nonselfadjoint Schrödinger operators. Reviewer: Elhadj Dahia (Bou Saâda) On the dynamics of the damped extensible beam 1D-equation https://zbmath.org/1517.47078 2023-09-22T14:21:46.120933Z "Lizama, Carlos" https://zbmath.org/authors/?q=ai:lizama.carlos "Murillo-Arcila, Marina" https://zbmath.org/authors/?q=ai:murillo-arcila.marina Summary: We show that the solutions of the linearized damped extensible beam equation exhibit a chaotic or stable behavior that depends on the distribution of the physical parameters of the equation. Such dynamical behavior is achieved in Herzog-like spaces. Our results provide new insights into the damped extensible beam equation by finding a critical parameter whose sign determines such qualitative properties. Markov approximations of the evolution of quantum systems https://zbmath.org/1517.47081 2023-09-22T14:21:46.120933Z "Gough, J." https://zbmath.org/authors/?q=ai:gough.john-e "Orlov, Yu. N." https://zbmath.org/authors/?q=ai:orlov.yurii-n "Sakbaev, V. Zh." https://zbmath.org/authors/?q=ai:sakbaev.vsevolod-zh "Smolyanov, O. G." https://zbmath.org/authors/?q=ai:smolyanov.oleg-georgievich Summary: The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini-Kossakowski-Sudarshan-Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators. Riesz projection and essential $$S$$-spectrum in quaternionic setting https://zbmath.org/1517.47122 2023-09-22T14:21:46.120933Z "Baloudi, Hatem" https://zbmath.org/authors/?q=ai:baloudi.hatem "Belgacem, Sayda" https://zbmath.org/authors/?q=ai:belgacem.sayda "Jeribi, Aref" https://zbmath.org/authors/?q=ai:jeribi.aref The authors study in this paper the Weyl and the essential $$S$$-spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the $$S$$-eigenvalue of finite type is both introduced and studied. It is also shown that the Weyl and the essential $$S$$-spectra do not contain eigenvalues of finite type. Further, the boundary of the Weyl $$S$$-spectrum and the particular case of the spectral theorem of the essential $$S$$-spectrum are also discussed. Reviewer: Dmitrii Legatiuk (Erfurt) Scattering theory for the Hodge Laplacian https://zbmath.org/1517.58008 2023-09-22T14:21:46.120933Z "Baumgarth, Robert" https://zbmath.org/authors/?q=ai:baumgarth.robert The Hodge Laplacian acting on differential k-forms carries important geometric and topological information about a manifold such as the spectrum of the operator. If the manifold $$M$$ is compact, then the spectrum consists of eigenvalues with finite multiplicity. If $$M$$ is non-compact, then the spectrum contains some absolutely continuous part. It might be asked, to what extent can the absolutely continuous part of the spectrum be controlled and under which assumptions about the geometry of the manifold. This is an interesting paper at 52 pages and 30 references. The paper addresses the natural question: can previous results concerning Laplacians acting on differential k-forms be extended to the setting of differential k-forms for two quasi- isometric Riemannian metrics? It is shown this can be done if the Weizenböck curvature endomorphisms reside in the Kato class and assuming an integral criterion only depending on a local upper bound of the heat kernel and certain explicitly given local curvature bounds. In addition, a necessary assumption will be a bound on a weight function measuring the first order deviation of the metrics in terms of the corresponding covariant derivatives. Reviewer: Paul Bracken (Edinburg) The mpEDMD algorithm for data-driven computations of measure-preserving dynamical systems https://zbmath.org/1517.65038 2023-09-22T14:21:46.120933Z "Colbrook, Matthew J." https://zbmath.org/authors/?q=ai:colbrook.matthew-j Summary: Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite dimensional, and computing their spectral information is a considerable challenge. We introduce \textit{measure-preserving extended dynamic mode decomposition} (\texttt{mpEDMD}), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. \texttt{mpEDMD} is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any preexisting dynamic mode decomposition (DMD)-type method, and with different types of data. We prove convergence of \texttt{mpEDMD} for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include the first convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate \texttt{mpEDMD} on a range of challenging examples, its increased robustness to noise compared to other DMD-type methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with Reynolds number $$> 6\times 10^4$$ and state-space dimension $$>10^5$$. A consistent description of the relativistic effects and three-body interactions in atomic nuclei https://zbmath.org/1517.81066 2023-09-22T14:21:46.120933Z "Yang, Y. L." https://zbmath.org/authors/?q=ai:yang.yunlei|yang.yulin|yang.yueli|yang.yalin|yang.yunlong.1|yang.yanli|yang.yaling|yang.yongliang|yang.yuliang|yang.yueling|yang.youlong|yang.yilong|yang.yinlong|yang.yuli|yang.yilu|yang.yanlong|yang.yuling|yang.yulu|yang.yunli|yang.yeong-ling|yang.yunle|yang.yong-li|yang.yulong|yang.yilin|yang.young-lyeol|yang.ya-lan|yang.yung-lieh|yang.yiling|yang.yanling|yang.yali|yang.yonglin "Zhao, P. W." https://zbmath.org/authors/?q=ai:zhao.peiwu|zhao.pengwei Summary: A microscopic relativistic Hamiltonian containing consistent relativistic and $$3N$$ potentials is, for the first time, constructed based on the leading-order covariant pionless effective field theory, and this Hamiltonian is solved by developing a new accurate relativistic \textit{ab initio} method with a novel symmetry-based artificial neural network for $$A \leq 4$$ nuclei. It is found that the relativistic effects overcome the energy collapse problem for $$^3\mathrm{H}$$ and $$^4\mathrm{He}$$ without promoting a repulsive three-nucleon interaction to leading order as in nonrelativistic calculations. To exactly reproduce the experimental ground-state energies, a three-nucleon interaction is needed and its interplay with the relativistic effects plays a crucial role. The presented results open the new avenue for a unified and consistent study on relativistic effects and many-body interactions in atomic nuclei, and would also help to achieve more accurate \textit{ab initio} calculations for nuclei. Constraints in the BV formalism: six-dimensional supersymmetry and its twists https://zbmath.org/1517.81079 2023-09-22T14:21:46.120933Z "Saberi, Ingmar" https://zbmath.org/authors/?q=ai:saberi.ingmar-a "Williams, Brian R." https://zbmath.org/authors/?q=ai:williams.brian-r Summary: We formulate the abelian six-dimensional $$\mathcal{N} = (2, 0)$$ theory perturbatively, in a generalization of the Batalin-Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an $$L_\infty$$ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the $$\mathcal{N} = (1, 0)$$ hypermultiplet, together with a degenerate holomorphic theory of holomorphic one-forms from the $$\mathcal{N} = (1, 0)$$ tensor multiplet, which can be seen to describe the infinitesimal intermediate Jacobian variety. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira-Spencer theory. Matching our formalism to five-dimensional Yang-Mills theory after reduction leads to some issues related to electric-magnetic duality; we offer some speculation on a nonperturbative resolution. Interacting massless infraparticles in 1+1 dimensions https://zbmath.org/1517.81086 2023-09-22T14:21:46.120933Z "Dybalski, Wojciech" https://zbmath.org/authors/?q=ai:dybalski.wojciech "Mund, Jens" https://zbmath.org/authors/?q=ai:mund.jens Summary: The Buchholz' scattering theory of waves in two dimensional massless models suggests a natural definition of a scattering amplitude. We compute such a scattering amplitude for charged infraparticles that live in the GNS representation of the $$2d$$ massless scalar free field and obtain a non-trivial result. It turns out that these excitations exchange phases, depending on their charges, when they collide. Dark energy star in gravity's rainbow https://zbmath.org/1517.83024 2023-09-22T14:21:46.120933Z "Bagheri Tudeshki, A." https://zbmath.org/authors/?q=ai:bagheri-tudeshki.a "Bordbar, G. H." https://zbmath.org/authors/?q=ai:bordbar.g-h "Eslam Panah, B." https://zbmath.org/authors/?q=ai:eslam-panah.b Summary: The concept of dark energy can be a candidate for preventing the gravitational collapse of compact objects to singularities. According to the usefulness of gravity's rainbow in UV completion of general relativity (by providing a new description of spacetime), it can be an excellent option to study the behavior of compact objects near phase transition regions. In this work, we obtain a modified Tolman-Openheimer-Volkof (TOV) equation for anisotropic dark energy as a fluid by solving the field equations in gravity's rainbow. Next, to compare the results with general relativity, we use a generalized Tolman-Matese-Whitman mass function to determine the physical quantities such as energy density, radial pressure, transverse pressure, gravity profile, and anisotropy factor of the dark energy star. We evaluate the junction condition and investigate the dynamical stability of dark energy star thin shell in gravity's rainbow. We also study the energy conditions for the interior region of this star. We show that the coefficients of gravity's rainbow can significantly affect this non-singular compact object and modify the model near the phase transition region. Noether charge formalism for Weyl transverse gravity https://zbmath.org/1517.83028 2023-09-22T14:21:46.120933Z "Alonso-Serrano, Ana" https://zbmath.org/authors/?q=ai:alonso-serrano.ana "Garay, Luis J." https://zbmath.org/authors/?q=ai:garay.luis-j "Liška, Marek" https://zbmath.org/authors/?q=ai:liska.marek Summary: Weyl transverse gravity (WTG) is a gravitational theory that is invariant under transverse diffeomorphisms and Weyl transformations. It is characterised by having the same classical solutions as general relativity while solving some of its issues with the cosmological constant. In this work, we first find the Noether currents and charges corresponding to local symmetries of WTG as well as a prescription for the symplectic form. We then employ these results to derive the first law of black hole mechanics in WTG (both in vacuum and in the presence of a perfect fluid), identifying the total energy, the total angular momentum, and the Wald entropy of black holes. We further obtain the first law and Smarr formula for Schwarzschild-anti-de Sitter and pure de Sitter spacetimes, discussing the contributions of the varying cosmological constant, which naturally appear in WTG. Lastly, we derive the first law of causal diamonds in vacuum. On the black hole acceleration in the C-metric space-time https://zbmath.org/1517.83033 2023-09-22T14:21:46.120933Z "Carneiro, F. L." https://zbmath.org/authors/?q=ai:carneiro.fernando-l-lobo-b "Ulhoa, S. C." https://zbmath.org/authors/?q=ai:ulhoa.sergio-c "Maluf, J. W." https://zbmath.org/authors/?q=ai:maluf.josee-w Summary: We consider the C-metric as a gravitational field configuration that describes an accelerating black hole in the presence of a semi-infinite cosmic string, along the accelerating direction. We adopt the expression for the gravitational energy-momentum developed in the teleparallel equivalent of general relativity (TEGR) and obtain an explanation for the acceleration of the black hole. The gravitational energy enclosed by surfaces of constant radius around the black hole is evaluated, and in particular, the energy contained within the gravitational horizon is obtained. This energy turns out to be proportional to the square root of the area of the horizon. We find that the gravitational energy of the semi-infinite cosmic string is negative and dominant for large values of the radius of integration. This negative energy explains the acceleration of the black hole that moves towards regions of lower gravitational energy along the string. Moduli space of stationary vacuum black holes from integrability https://zbmath.org/1517.83046 2023-09-22T14:21:46.120933Z "Lucietti, James" https://zbmath.org/authors/?q=ai:lucietti.james "Tomlinson, Fred" https://zbmath.org/authors/?q=ai:tomlinson.fred Summary: We consider the classification of asymptotically flat, stationary, vacuum black hole spacetimes in four and five dimensions, that admit one and two commuting axial Killing fields respectively. It is well known that the Einstein equations reduce to a harmonic map on the two-dimensional orbit space, which itself arises as the integrability condition for a linear system of spectral equations. We integrate the Belinski-Zakharov spectral equations along the boundary of the orbit space and use this to fully determine the metric and associated Ernst and twist potentials on the axes and horizons. This is sufficient to derive the moduli space of solutions that are free of conical singularities on the axes, for any given rod structure. As an illustration of this method we obtain constructive uniqueness proofs for the Kerr and Myers-Perry black holes and the known doubly spinning black rings. A new class of regular black hole solutions with quasi-localized sources of matter in $$(2 + 1)$$ dimensions https://zbmath.org/1517.83047 2023-09-22T14:21:46.120933Z "Maluf, R. V." https://zbmath.org/authors/?q=ai:maluf.roberto-v "Muniz, C. R." https://zbmath.org/authors/?q=ai:muniz.celio-r "Santos, A. C. L." https://zbmath.org/authors/?q=ai:santos.a-c-l "Estrada, Milko" https://zbmath.org/authors/?q=ai:estrada.milko Summary: This paper investigates a new class of regular black hole solutions in $$(2 + 1)$$-dimensions by introducing a generalization of the quasi-localized matter model proposed by \textit{M. Estrada} and \textit{F. Tello-Ortiz} [Europhys. Lett. 135, No. 2, Article ID 20001, 6 p. (2021; \url{doi:10.1209/0295-5075/ac0ed0})]. Initially, we try to physically interpret the matter source encoded in the energy-momentum tensor as originating from nonlinear electrodynamics. We show, however, that the required conditions for the quasi-locality of the energy density are incompatible with the expected behavior of nonlinear electrodynamics, which must tend to Maxwell's theory on the asymptotic limit. Despite this, we propose a generalization for the quasi-localized energy density that encompasses the existing models in the literature and allows us to obtain a class of regular black hole solutions exhibiting remarkable features on the event horizons and their thermodynamic properties. Furthermore, since the usual version of the first law of thermodynamics, due to the presence of the matter fields, leads to incorrect values of entropy and thermodynamics volume for regular black holes, we propose a new version of the first law for regular black holes. Torsion-induced chiral magnetic current in equilibrium https://zbmath.org/1517.83069 2023-09-22T14:21:46.120933Z "Amitani, Tatsuya" https://zbmath.org/authors/?q=ai:amitani.tatsuya "Nishida, Yusuke" https://zbmath.org/authors/?q=ai:nishida.yusuke Summary: We study equilibrium transport properties of massless Dirac fermions at finite temperature and chemical potential in spacetime accompanied by torsion, which in four dimensions couples with Dirac fermions as an axial gauge field. In particular, we compute the current density at the linear order in the torsion as well as in an external magnetic field with the Pauli-Villars regulatization, finding that an equilibrium current akin to the chiral magnetic current is locally induced. Such torsion can be realized in condensed matter systems along a screw dislocation line, around which localized and extended current distributions are predicted so as to be relevant to Dirac and Weyl semimetals. Furthermore, we compute the current density at the linear order in the torsion as well as in a Weyl node separation, which turns out to vanish in spite of being allowed from the symmetry perspective. Contrasts of our findings with torsion-induced currents from previous work are also discussed. Evaluating the spin-0 particle in Gurses rainbow universe https://zbmath.org/1517.83079 2023-09-22T14:21:46.120933Z "Kangal, E. E." https://zbmath.org/authors/?q=ai:kangal.evrim-ersin Summary: In this study, we solved the Klein-Gordon's oscillator equation in Gurses rainbow universe. Subsequently, the energy quantization and associated wave function are obtained with the help of the Nikiforov-Uvarov technique. In the final stage, we compare graphically the rainbow effect with General Relativity (GR) one. According to the obtained result, the occurrence of asymmetrical breaking between positive and negative energy states has been observed in the rainbow scenario.