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Abstract: Abstract We introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces \(L_p\) , \(p \notin 2{\mathbb {N}}+4\) are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive \(L_p\) -spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of \(L_p\) by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of \(L_p\) . PubDate: 2024-07-12

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Abstract: Abstract Let \(0<p<\infty , \alpha >-1,\) and \(\beta ,\gamma \in {\mathbb {R}}.\) Let \(\mu \) be a finite positive Borel measure on the unit disk \({\mathbb {D}}.\) The Zygmund space \(L^{p,\beta }(d\mu )\) consists of all measurable functions f on \({\mathbb {D}}\) such that \( f ^p\log ^\beta (e+ f )\in L^1(d\mu )\) and the Bergman–Zygmund space \(A^{p,\beta }_{\alpha }\) is the set of all analytic functions in \(L^{p,\beta }(dA_\alpha ),\) where \(dA_\alpha =c_\alpha (1- z ^2)^\alpha dA.\) We prove an interpolation theorem for the Zygmund space assuming the weak type estimates on the Zygmund spaces themselves at the end points rather than the weak \(L^p-L^q\) type estimates at the end points. We show that the Bergman–Zygmund space is equal to the \(\log ^\beta (e/(1- z )) dA_\alpha (z)\) weighted Bergman space as a set and characterize the bounded and compact Carleson measure \(\mu \) from \(A^{p,\beta }_{\alpha }\) into \(A^{p,\gamma }(d\mu ),\) respectively. The Carleson measure characterizations are of the same type for any pairs of \((\beta , \gamma )\) whether \(\beta <\gamma \) or \(\gamma \le \beta .\) PubDate: 2024-07-08

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Abstract: Abstract We establish that the Volterra-type integral operator \(J_b\) on the Hardy spaces \(H^p\) of the unit ball \({\mathbb {B}}^n\) exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and \(\ell ^p\) -singularity of \(J_b\) are equivalent on \(H^p\) for any \(1 \le p < \infty \) . Moreover, we show that the operator \(J_b\) acting on \(H^p\) cannot fix an isomorphic copy of \(\ell ^2\) when \(p \ne 2.\) PubDate: 2024-07-02

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Abstract: Abstract Let (X, d) be a compact metric space and \({\mathcal {A}}\) be a commutative and semisimple Banach algebra. Some of our recent works are related to the several \(\mathrm BSE\) concepts of the vector-valued Lipschitz algebra \(\textrm{Lip}(X,{\mathcal {A}})\) . In this paper as the main purpose, we verify the \(\mathrm BED\) property for \(\textrm{Lip}(X,{\mathcal {A}})\) , which is actually different from the \(\mathrm BSE\) feature. We first prove as an elementary result that \(\textrm{Lip}(X,{\mathcal {A}})\) is regular if and only if \({\mathcal {A}}\) is so. Then we prove that \({\mathcal {A}}\) is a \(\mathrm BED\) algebra, whenever \(\textrm{Lip}(X,{\mathcal {A}})\) is so. Afterwards, we verify the converse of this statement. Indeed, we prove that if \({\mathcal {A}}\) is a \(\mathrm BED\) algebra then \(C^{0}_{\textrm{BSE}}(\Delta (\textrm{Lip}(X,{\mathcal {A}})))\subseteq \widehat{\textrm{Lip}(X,{\mathcal {A}})}\) and \(\widehat{\textrm{Lip}X\otimes {\mathcal {A}}}\subseteq C^{0}_{\textrm{BSE}}(\Delta (\textrm{Lip}(X,{\mathcal {A}}))).\) It follows that if \(\textrm{Lip}X\otimes {\mathcal {A}}\) is dense in \(\textrm{Lip}(X,{\mathcal {A}})\) then \(\textrm{Lip}(X,{\mathcal {A}})\) is a \(\mathrm BED\) algebra, provided that \({\mathcal {A}}\) is so. Moreover, we conclude that the necessary and sufficient condition for the unital and in particular finite dimensional Banach algebra \({\mathcal {A}}\) , to be a \(\mathrm BED\) algebra is that \(\textrm{Lip}(X,{\mathcal {A}})\) is a \(\mathrm BED\) algebra. Finally, regarding to some known results which disapproves the \(\mathrm BSE\) property for \(\textrm{lip}_{\alpha }(X,{\mathcal {A}})\) \((0<\alpha <1\) ), we show that for any commutative and semisimple Banach algebra \({\mathcal {A}}\) with PubDate: 2024-07-01

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Abstract: Abstract The main aim of the paper is to investigate some sufficient conditions which guarantee the convergence of sequences of positive linear operators towards composition operators within the framework of function spaces defined on a metric space. Among other things, the adopted approach allows to obtain a unifying reassessment of two milestones of the approximation theory by positive linear operators, namely, Korovkin’s theorem and Feller’s theorem together with some new extensions of them to the more general case where the limit operator is a composition operator. Some applications are shown and, among them, the convergence of Bernstein–Schnabl operator is enlightened in the framework of Banach spaces. PubDate: 2024-06-28

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Abstract: Abstract In this article, some new martingale inequalities in the framework of Orlicz–Karamata modular spaces are discussed. More precisely, we establish modular inequalities associated with Orlicz functions and slowly varying functions. The results obtained herein can weaken the restrictive condition that the slowly varying function b is nondecreasing in (Math Nachr 291(8–9):1450–1462, 2018). PubDate: 2024-06-26

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Abstract: Abstract We characterize the sequences of complex numbers \((z_{n})_{n \in \mathbb {N}}\) and the locally complete (DF)-spaces E such that for each \((e_{n})_{n \in \mathbb {N}} \in E^\mathbb {N}\) there exists an E-valued function \(\textbf{f}\) on \((0,\infty )\) (satisfying a mild regularity condition) such that $$\begin{aligned} \int _{0}^{\infty } t^{z_{n}} \textbf{f}(t) dt = e_{n}, \qquad \forall n \in \mathbb {N}, \end{aligned}$$ where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution \(\textbf{f}\) that is smooth on \((0,\infty )\) and satisfies certain optimal growth bounds near 0 and \(\infty \) . The scalar-valued case \((E = \mathbb {C})\) was treated by Durán (Math Nachr 158:175–194, 1992). Our work is based upon his result. PubDate: 2024-06-25

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Abstract: Abstract The spectral problem for the Schrödinger operator with a magnetic field (the Landau operator) on the flat Möbius strip is considered. The model of the flat Möbius strip is based on gluing the rectangles. The spectrum and eigenfunctions for the model operator are described. The results are compared with the flat cylinder case. “Surgery” of the flat Möbius strip (by longitudinal cutting) is discussed. PubDate: 2024-06-11

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Abstract: Abstract This paper addresses the link between the characterization of spectral sets for orthogonal Fourier bases in \({\mathbb {Z}}_n^2\) and for discrete orthogonal Gabor bases in \({\mathbb {Z}}_n^2.\) For Fourier bases there is a well known conjecture (i.e., the Fuglede conjecture) which states that a set is spectral if and only if it is a tile. This conjecture has been disproved for \(d\ge 3\) (d for dimension) and remains open for \(d=1,2.\) A similar but stricter characterization for discrete orthogonal Gabor bases is conjectured here, which states that the support set shall not only be tiling but also either a subgroup of order n (i.e., a Lagrangian) or a tiling complement of such a subgroup. The additional requirement comes from restrictions on the window vector. The author has established this statement (in both directions) before for n being a prime number, the purpose of this paper is to extend this result to n being a prime square. As opposed to possible false first impressions, this is not a simple extension of the prime case, and actually relies heavily on several new techniques. PubDate: 2024-06-08

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Abstract: Abstract It is known that the solutions of the fractional Cauchy problems on Banach spaces are given by the corresponding fractional resolvent families. To study the asymptotic behaviors of the solutions, we give a sufficient and necessary condition for the uniform stability of fractional resolvent families in terms of the behavior of the resolvents of their generators on Hilbert spaces, which generalizes the classical Gearhart–Prüss theorem. And we completely characterize the exponential growth bound of fractional resolvent families on Hilbert spaces. On Banach spaces, a necessary condition for the uniform stability is also given. Moreover, by using a functional equation for fractional resolvent families, we show a uniformly stable \(\alpha \) -times resolvent family with generator A will approach to \(\frac{-A^{-1}}{\Gamma (1-\alpha )t^\alpha }\) as \(t \rightarrow +\infty \) , and the optimal convergence rates are derived. PubDate: 2024-06-07

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Abstract: Abstract We prove that for every countable ordinal \(\xi \) , the Tsirelson’s space \(T_\xi \) of order \(\xi \) , is naturally, i.e., via the identity, 3-isomorphic to its modified version. For the first step, we prove that the Schreier family \(\mathcal {S}_\xi \) is the same as its modified version \( \mathcal {S}^M_\xi \) , thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on \(T_\xi \) has \(2^{{\mathfrak {c}}}\) closed ideals. PubDate: 2024-06-03

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Abstract: Abstract In this article, we discuss the applications of martingale Hardy Orlicz–Lorentz–Karamata spaces in Fourier analysis. More precisely, we show that the partial sums of the Walsh–Fourier series converge to the function in norm if \(f\in L_{\Phi ,q,b}\) with \(1<p_-\le p_+<\infty \) . The equivalence of maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces is presented. The Fejér summability method is also studied and it is proved that the maximal Fejér operator is bounded from martingale Hardy Orlicz–Lorentz–Karamata spaces to Orlicz–Lorentz–Karamata spaces. As a consequence, we obtain conclusions about almost everywhere and norm convergence of Fejér means. PubDate: 2024-06-03

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Abstract: Abstract In this article, we discuss the relationship between Birkhoff–James orthogonality of elementary tensors in the space \(L^{p}(\mu )\otimes ^{\Delta _{p}}X,\; (1\le p<\infty )\) with the individual elements in their respective spaces, where X is a Banach space whose norm is Fr \(\acute{e}chet\) differentiable and \(\Delta _{p}\) is the natural norm induced by \(L^{p}(\mu ,X)\) . In order to study the said relationship, we first provide some characterizations of Birkhoff–James orthogonality of elements in the Lebesgue-Bochner space \(L^{p}(\mu ,X)\) . PubDate: 2024-05-28

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Abstract: Abstract Let \(\mathcal {B(H)}\) be the collection of bounded linear operators on a complex separable Hilbert space \(\mathcal {H}\) . For \(T\in \mathcal {B(H)}\) , its numerical range and maximal numerical range are denoted by W(T) and \(W_0(T)\) , respectively. First, we give in this paper a characterization of the maximal numerical range and, as applications, we determine maximal numerical ranges of weighted shifts, partial isometries, the Volterra integral operator and classical Toeplitz operators. Second, we study the universality of maximal numerical ranges, showing that any nonempty bounded convex closed subset of \(\mathbb {C}\) is the maximal numerical range of some operator. Finally, we discuss the relations among the numerical range, the maximal numerical range and the spectrum. It is shown that the collection of those operators T with \(W_0(T)\cap \sigma (T)=\emptyset \) is a nonempty open subset of \(\mathcal {B(H)}\) precisely when \(\dim \mathcal {H}>1\) , and is dense precisely when \(1<\dim \mathcal {H}<\infty \) . We also show that those operators T with \(W_0(T)= W(T)\) constitute a nowhere dense subset of \(\mathcal {B(H)}\) precisely when \(\dim \mathcal {H}>1\) PubDate: 2024-05-27

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Abstract: Abstract In this paper, we consider a class of generalized closed linear manifolds in a nonseparable Hilbert space H, which is closely related to the generalized Fredholm theory. We first investigate properties of the set \({\mathcal {B}}_{\vartriangleleft }=\{T\in {\mathcal {M}}:\overline{T(H)}\subset A(H)\) for some \(A\in {\mathcal {B}}\},\) where \({\mathcal {B}}\) is a \(C^*\) -subalgebra of a von Neumann algebra \({\mathcal {M}}\) . It is proved that a selfadjoint \({\mathcal {B}}_{\vartriangleleft }\) is always an ideal in \({\mathcal {M}}\) . In a type \(\textrm{II}_\infty \) factor, we show that there exists a tracial weight (whose range containing infinite cardinals) such that two projections are equivalent if and only if they have the same tracial weight, which leads to a complete characterization of such selfadjoint \({\mathcal {B}}_{\vartriangleleft }\) when \({\mathcal {M}}\) is a factor. Then we introduce the concept of closed manifolds with respect to a pair of C*-algebras and study some properties. Finally, when m is an infinite cardinal, as a special important case we focus on m-closed subspaces and operators which preserve m-closed subspaces. It is proved that these operators are either of rank less than m, or the generalized left semi-Fredholm operators. PubDate: 2024-05-27

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Abstract: Abstract Let T be a multilinear Calderón–Zygmund operator of type \(\omega \) . \(T_{\vec {b},S}\) is the generalized commutator of T, which generalizes several commutators that already existed. It is shown in this paper that the weak and strong type quantitative weighted estimates for \(T_{\vec {b},S}\) when \(\vec {b}=\{b_i\}_{i=1}^{\infty }\) belongs to exponential oscillation spaces and Lipschitz spaces, respectively. As applications, we obtain the multiple weighted norm inequalities for the generalized commutators of bilinear pseudo-differential operators and paraproducts with mild regularity. PubDate: 2024-05-27

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Abstract: Abstract First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d- \(\sigma \) -stability in a random metric space can be regarded as a special case of the notion of \(\sigma \) -stability in a random normed module; as another application we give the final version of the characterization for a d- \(\sigma \) -stable random metric space to be stably compact. Second, we prove that an \(L^{p}\) -normed \(L^{\infty }\) -module is exactly generated by a complete random normed module so that the gluing property of an \(L^{p}\) -normed \(L^{\infty }\) -module can be derived from the \(\sigma \) -stability of the generating random normed module, as applications the direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is \((\varepsilon ,\lambda )\) -complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d- \(\sigma \) -stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation. PubDate: 2024-05-24

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Abstract: Abstract Let G be a locally compact group and let \(A_\Phi (G)\) be the Orlicz version of the Figà–Talamanca Herz algebra of G associated with a Young function \(\Phi .\) We show that if \(A_\Phi (G)\) is Arens regular, then G is discrete. We further explore the Arens regularity of \(A_\Phi (G)\) when the underlying group G is discrete. In the running, we also show that \(A_\Phi (G)\) is finite dimensional if and only if G is finite. Further, for amenable groups, we show that \(A_\Phi (G)\) is reflexive if and only if G is finite, under the assumption that the associated Young function \(\Phi \) satisfies the MA condition. PubDate: 2024-05-13

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Abstract: Abstract This paper deals mainly with some aspects of the adjointable operators on Hilbert \(C^*\) -modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert \(C^*\) -modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace M of certain Hilbert space K and an operator \(T\in {\mathbb {B}}(K)\) such that T is (M, M)-weakly complementable, whereas T fails to be (M, M)-complementable. The solvability of the equation $$\begin{aligned} A:B=X^*AX+(I-X)^*B(I-X) \quad \big (X\in {\mathbb {B}}(H)\big ) \end{aligned}$$ is also dealt with in the Hilbert space case, where \(A,B\in {\mathbb {B}}(H)\) are two general positive operators, and A : B denotes their parallel sum. Among other things, it is shown that there exist certain positive operators A and B on the Hilbert space \(\ell ^2({\mathbb {N}})\oplus \ell ^2({\mathbb {N}})\) such that the above equation has no solution. PubDate: 2024-05-11