Recent zbMATH articles in MSC 32Ahttps://zbmath.org/atom/cc/32A2023-05-08T18:47:08.967005ZWerkzeugLidstone interpolation. III: Several variableshttps://zbmath.org/1507.410122023-05-08T18:47:08.967005Z"Waldschmidt, Michel"https://zbmath.org/authors/?q=ai:waldschmidt.michelUnivariate polynomials can be identified uniquely by Hermite interpolation conditions at two points in the complex plane, the arguments zero and one, say. The order of these Hermite conditions has to be sufficiently large, and it is suitable to demand the conditions at the two points with only even-order derivatives. In this paper, this theory is generalised to \(n\) dimensions, and the points at 0 and 1 are replaced by suitable unit vectors in \(n\) variables. A canonical representation using this Hermite interpolation information is an expansion into linear combinations of so-called Lidstone polynomials that replace the well-known Lagrange functions in Lagrange and Hermite interpolation of the classical form.
Reviewer: Martin D. Buhmann (Gießen)A survey on the Arveson-Douglas conjecturehttps://zbmath.org/1507.470202023-05-08T18:47:08.967005Z"Guo, Kunyu"https://zbmath.org/authors/?q=ai:guo.kunyu"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.21The Arveson-Douglas conjecture refers to a set of conjectures involving essential normality of submodules and quotient modules of analytic function spaces.
In this survey, the authors describe several proven results over the past 50 years related to this conjecture. These results are complemented with questions related to the discussed topics. These topics include geometric invariants for row contractions, index theorems, holomorphic extension theorems, principal submodules, decomposition modules, the geometric Arveson-Douglas conjecture, decomposition of submodules and quotient modules, and quotient modules over the polydisc, among others.
For the entire collection see [Zbl 1455.47001].
Reviewer: Joan Fàbrega (Barcelona)Ranks of commutators for a class of truncated Toeplitz operatorshttps://zbmath.org/1507.470682023-05-08T18:47:08.967005Z"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.4"Lee, Young Joo"https://zbmath.org/authors/?q=ai:lee.young-joo"Zhao, Yile"https://zbmath.org/authors/?q=ai:zhao.yileSummary: We consider truncated Toeplitz operators acting on infinite dimensional model spaces. We then describe the kernels and ranks of commutators of truncated Toeplitz operators with symbols induced by certain inner functions. Our results generalize recent results of \textit{Y. Chen} et al. [Oper. Matrices 15, No. 1, 85--103 (2021; Zbl 1489.47048)] to infinite dimensional model spaces.Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbolshttps://zbmath.org/1507.471232023-05-08T18:47:08.967005Z"Rodriguez, Miguel Angel Rodriguez"https://zbmath.org/authors/?q=ai:rodriguez.miguel-angel-rodriguezSummary: Let \(D_3\) be the three-dimensional Siegel domain and \({\mathcal{A}}_\lambda^2(D_3)\) the weighted Bergman space with weight parameter \(\lambda >-1\). In the present paper, we analyse the commutative (not \(C^*)\) Banach algebra \({\mathcal{T}}(\lambda)\) generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on \({\mathcal{A}}_\lambda^2(D_3)\). We remark that \({\mathcal{T}}(\lambda)\) is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed.Algebra generated by Toeplitz operators with \(\mathbb{T} \)-invariant symbolshttps://zbmath.org/1507.471242023-05-08T18:47:08.967005Z"Vasilevski, Nikolai"https://zbmath.org/authors/?q=ai:vasilevski.nikolai-lSummary: We study the structure of the \(C^*\)-algebras generated by Toeplitz operators acting on the weighted Bergman space \(\mathcal{A}^2_{\lambda}(\mathbb{B}^2)\) on the two-dimensional unit ball, whose symbols are invariant under the action of the group \(\mathbb{T} \). We consider three principally different basic cases of its action \(t:\,(z_1,z_2) \mapsto (tz_1,t^{k_2}z_2)\), with \(k_2=1,0,-1\). The properties of the corresponding Toeplitz operators as well as the structure of the \(C^*\)-algebra generated by them turn out to be drastically different for these three cases.Essential commutants on strongly pseudo-convex domainshttps://zbmath.org/1507.471252023-05-08T18:47:08.967005Z"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.21"Xia, Jingbo"https://zbmath.org/authors/?q=ai:xia.jingboSummary: Consider a bounded strongly pseudo-convex domain \(\Omega\) with smooth boundary in \(\mathbf{C}^n\). Let \(\mathcal{T}\) be the Toeplitz algebra on the Bergman space \(L_a^2({\Omega})\). That is, \( \mathcal{T}\) is the \(C^\ast \)-algebra generated by the Toeplitz operators \(\{ T_f : f \in L^\infty({\Omega}) \} \). Extending the work [the second author, J. Funct. Anal. 272, No. 12, 5191--5217 (2017; Zbl 06714269); J. Funct. Anal. 274, No. 6, 1631--1656 (2018; Zbl 1394.46044)] in the special case of the unit ball, we show that on any such \(\Omega\), \(\mathcal{T}\) and \(\{ T_f : f \in \mathrm{VO}_{\mathrm{bdd}} \} + \mathcal{K}\) are essential commutants of each other, where \(\mathcal{K}\) is the collection of compact operators on \(L_a^2({\Omega})\). On a general \(\Omega\) considered in this paper, the proofs require many new ideas and techniques. These same techniques also enable us to show that, for \(A \in \mathcal{T} \), if \(\langle A k_z, k_z \rangle \to 0\) as \(z \to \partial {\Omega} \), then \(A\) is a compact operator.Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMshttps://zbmath.org/1507.471282023-05-08T18:47:08.967005Z"Ghiloni, Riccardo"https://zbmath.org/authors/?q=ai:ghiloni.riccardo"Moretti, Valter"https://zbmath.org/authors/?q=ai:moretti.valter"Perotti, Alessandro"https://zbmath.org/authors/?q=ai:perotti.alessandroSummary: The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann's foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator [\textit{F. Colombo} et al., Electron. Res. Announc. Math. Sci. 14, 60--68 (2007; Zbl 1137.47014)]. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A~problem of the quaternionic formulation is the description of composite quantum systems in the absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A~notion with this property, called \textit{intertwining quaternionic PVM}, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover, in particular, the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.Non-autonomous rough semilinear PDEs and the multiplicative sewing lemmahttps://zbmath.org/1507.601482023-05-08T18:47:08.967005Z"Gerasimovičs, Andris"https://zbmath.org/authors/?q=ai:gerasimovics.andris"Hocquet, Antoine"https://zbmath.org/authors/?q=ai:hocquet.antoine"Nilssen, Torstein"https://zbmath.org/authors/?q=ai:nilssen.torstein-kSummary: We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form \(d u_t - L_t u_t d t = N( u_t) d t + \sum_{i = 1}^d F_i( u_t) d \mathbf{X}_t^i\) where \(( L_t )_{t \in [ 0 , T ]}\) is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while \(\mathbf{X} \equiv(X, \mathbb{X})\) is a two-step (non-necessarily geometric) rough path of Hölder regularity \(\gamma > 1 / 3\). Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path \(\mathbf{X}\)). As a technical tool, we introduce a version of the multiplicative sewing lemma, which allows to construct the so called product integrals in infinite dimensions. We later use it to construct a semigroup analogue for the non-autonomous linear PDEs as well as show how to deduce the semigroup version of the usual sewing lemma from it.