Recent zbMATH articles in MSC 46Ahttps://zbmath.org/atom/cc/46A2023-09-22T14:21:46.120933ZWerkzeugExtendability of continuous quasiconvex functions from subspaceshttps://zbmath.org/1517.260092023-09-22T14:21:46.120933Z"De Bernardi, Carlo Alberto"https://zbmath.org/authors/?q=ai:de-bernardi.carlo-alberto"Veselý, Libor"https://zbmath.org/authors/?q=ai:vesely.liborSummary: Let \(Y\) be a subspace of a topological vector space \(X\), and \(A\subset X\) an open convex set that intersects \(Y\). We say that the property \((QE)\) [property \((CE)]\) holds if every continuous quasiconvex [continuous convex] function on \(A\cap Y\) admits a continuous quasiconvex [continuous convex] extension defined on \(A\). We study relations between \((QE)\) and \((CE)\) properties, proving that \((QE)\) always implies \((CE)\) and that, under suitable hypotheses (satisfied for example if \(X\) is a normed space and \(Y\) is a closed subspace of \(X)\), the two properties are equivalent. By combining the previous implications between \((QE)\) and \((CE)\) properties with known results about the property \((CE)\), we obtain some new positive results about the extension of quasiconvex continuous functions. In particular, we generalize the results contained in [9] to the infinite-dimensional separable case. Moreover, we also immediately obtain existence of examples in which \((QE)\) does not hold.On unbounded order continuous operatorshttps://zbmath.org/1517.460022023-09-22T14:21:46.120933Z"Turan, Bahri"https://zbmath.org/authors/?q=ai:turan.bahri"Altin, Birol"https://zbmath.org/authors/?q=ai:altin.birol"Gürkök, Hüma"https://zbmath.org/authors/?q=ai:gurkok.humaSummary: Let \(U\) and \(V\) be two Archimedean Riesz spaces. An operator \(S: U\rightarrow V\) is said to be unbounded order continuous (\(uo\)-continuous), if \(r_\alpha\overset{uo}{\rightarrow}0\) in \(U\) implies \(Sr_\alpha\overset{uo}{\rightarrow}0\) in \(V\). In this paper, we give some properties of the \(uo\)-continuous dual \(U_{uo}^\sim\) of \(U\). We show that a nonzero linear functional \(f\) on \(U\) is \(uo\)-continuous if and only if \(f\) is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the \(uo\)-continuous dual \(U_{uo}^\sim\). In general, by giving an example that the \(uo\)-continuous dual \(U_{uo}^\sim\) is not a band in \(U^\sim\), we obtain the conditions for the \(uo\)-continuous dual of a Banach lattice \(U\) to be a band in \(U^\sim\). Then, we examine the properties of \(uo\)-continuous operators. We show that \(S\) is an order continuous operator if and only if \(S\) is an unbounded order continuous operator when \(S\) is a lattice homomorphism between two Riesz spaces \(U\) and \(V\). Finally, we proved that if an order bounded operator \(S:U\rightarrow V\) between Archimedean Riesz space \(U\) and atomic Dedekind complete Riesz space \(V\) is \(uo\)-continuous, then \(|S|\) is \(uo\)-continuous.The intrinsic core and minimal faces of convex sets in general vector spaceshttps://zbmath.org/1517.460032023-09-22T14:21:46.120933Z"Millán, R. Díaz"https://zbmath.org/authors/?q=ai:millan.reinier-diaz"Roshchina, Vera"https://zbmath.org/authors/?q=ai:roshchina.veraThe relative interior of a convex set in a finite-dimensional real vector space is the interior of this convex set relative to its affine hull; a direct generalization of this notion to real vector spaces is the intrinsic core, introduced in [\textit{V. L. Klee jun.}, Duke Math. J. 18, 443--466 (1951; Zbl 0042.36201)] and developed in [\textit{R. B. Holmes}, Geometric functional analysis and its applications. New York-Heidelberg-Berlin: Springer-Verlag (1975; Zbl 0336.46001)]. The intrinsic core appears in the literature under different names, including the set of relatively absorbing points, the pseudo-relative interior, and the set of inner points.
The authors discuss four equivalent definitions of the intrinsic core, via line segments, the cone of feasible directions, as a core with respect to the affine hull, and in terms of minimal faces. The differences in notation and discrepancies in implicit assumptions used by different authors are discussed in detail. The paper focuses on the interplay between the intrinsic core and facial structure in the restricted setting of real vector spaces without topological structure. Well-known results and examples discussed here are scattered in the literature, and it may be difficult to find neat references for widely known statements. Besides, the authors are able to provide some new insights and discuss several examples of interesting infinite-dimensional convex sets from the perspective of facial structure: for instance, they construct uncountable chains of faces of the Hilbert cube.
Reviewer: Ioan Raşa (Cluj-Napoca)Dense quasi-free subalgebras of the Toeplitz algebrahttps://zbmath.org/1517.460352023-09-22T14:21:46.120933Z"Pirkovskii, A. Yu."https://zbmath.org/authors/?q=ai:pirkovskii.alexei-yulevichSummary: We introduce a family of dense subalgebras of the Toeplitz algebra and give conditions under which our algebras are quasi-free. As a corollary, we show that the smooth Toeplitz algebra introduced by Cuntz is quasi-free.The spectrum and fine spectrum of generalized Rhaly-Cesàro matrices on \(c_0\ \mathrm{and}\ c\)https://zbmath.org/1517.470592023-09-22T14:21:46.120933Z"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.caglaSummary: 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.Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaceshttps://zbmath.org/1517.470862023-09-22T14:21:46.120933Z"Salame, Khadime"https://zbmath.org/authors/?q=ai:salame.khadimeLet WAP\((S)\) denote the Banach algebra of weakly almost periodic functions on a given semitopological semigroup \(S\). During an International Conference on Fixed Point Theory which took place in Halifax in 1975, Anthony To-Ming Lau raised the question whether or not the left amenability property of WAP\((S)\) is equivalent to the existence of a common fixed point of any separately weakly continuous and weakly quasi-equicontinuous nonexpansive action of \(S\) on a weakly compact convex subset of a separated locally convex space. This remained an open problem for almost 50 years. In the present paper, the author gives an affirmative answer.
Reviewer: Jürgen Appell (Würzburg)Feral dual spaces and (strongly) distinguished spaces \(C(X)\)https://zbmath.org/1517.540042023-09-22T14:21:46.120933Z"Kąkol, Jerzy"https://zbmath.org/authors/?q=ai:kakol.jerzy"Śliwa, Wiesław"https://zbmath.org/authors/?q=ai:sliwa.wieslawThe paper is dedicated to the study of locally convex spaces, in particular of spaces of the form \(C_p(X)\) and \(C_k(X)\). Here, for a Tychonoff space \(X\), \(C_p(X)\) and \(C_k(X)\) denote the vector space \(C(X)\) of all real-valued continuous functions on \(X\) endowed with the pointwise topology and the compact-open topology, respectively. Recall that a locally convex space \(E\) is \textit{distinguished} if its strong dual \(E_\beta'\) is barrelled (namely, if every absolutely convex, absorbing, and closed subset of \(E_\beta'\) is a neighbourhood of the origin). Section 2 of the article has partially a survey flavour and collects several equivalent conditions for \(C_p(X)\) being distinguished (Theorem 6). Even in the case of known characterisations, a self-contained (and often new) proof is given. Among the equivalent properties is the fact that the strong dual of \(C_p(X)\) carries the finest locally convex topology. This motivates the notion of strongly distinguished locally convex space (Definition 1): a locally convex space \(E\) is \textit{strongly distinguished} if its strong dual \(E_\beta'\) carries the finest locally convex topology (equivalently, if every absolutely convex and absorbing subset of \(E_\beta'\) is a neighbourhood of the origin).
The second part of the article is dedicated to feral locally convex spaces: a locally convex space \(E\) is \textit{feral} if bounded sets in \(E\) are finite-dimensional. A result from [\textit{J. C. Ferrando} et al., Funct. Approximatio, Comment. Math. 50, No. 2, 389--399 (2014; Zbl 1319.46002)] states that the strong dual of \(C_p(X)\) is feral, for each Tychonoff space \(X\). Generalising this result, the authors give a characterisation of those locally convex spaces whose strong dual is feral (Theorem 7). Among the applications of this result is the following neat characterisation of spaces \(C_k(X)\) with feral strong dual (Theorem 2): the strong dual of \(C_k(X)\) is feral if and only if every compact subset of \(X\) is finite. Together with results from Section 2, this also yields a characterisation of those \(C_k(X)\) spaces that are strongly distinguished (Corollary 2).
Reviewer: Tommaso Russo (Innsbruck)Infinite-dimensional vector optimization and a separation theoremhttps://zbmath.org/1517.901302023-09-22T14:21:46.120933Z"Amahroq, Tijani"https://zbmath.org/authors/?q=ai:amahroq.tijani"Oussarhan, Abdessamad"https://zbmath.org/authors/?q=ai:oussarhan.abdessamadNecessary and sufficient optimality conditions for relative minimal solutions of vector optimization problems are studied. The authors establish for this purpose a new separation theorem, which encompasses both finite- and infinite-dimensional classical separation theorems. The new separation theorem involves the notion of compactly epi-Lipschitz sets in normed vector spaces introduced probably for the first time in [\textit{J. M. Borwein} and \textit{H. M. Strojwas}, Nonlinear Anal., Theory Methods Appl. 9, 1347--1366 (1985; Zbl 0613.49016)]. Using the separation theorem, necessary and sufficient conditions for quasi-relative solutions in terms of Karush-Kuhn-Tucker multipliers are derived. The general separation theorem is used further to obtain Fritz-John multipliers for vector optimization problems. In the last section of the paper, the authors use the obtained Fritz-John multipliers to prove necessary optimality conditions for vector optimization problems.
Reviewer: Karel Zimmermann (Praha)