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Abstract: Abstract Let E, F and G be Banach lattices. \({\mathscr {A}}_{G}(E,F)\) is the collection of all bounded operators from E to F, that \(TU:G\rightarrow F\) is regular for every bounded linear operator \(U:G\rightarrow E\) . The operator class \({\mathscr {A}}_{G}(E,F)\) has studied. A result of Yanovskii show that \(T:E\rightarrow F\in {\mathscr {A}}_{c_0}(E,F^{\prime \prime })\) if and only if T is weak serially summing operator i.e. the series [Yanovskii, L.P (Math. J. 20(2); 287–292, 1979)] \(\sum \limits _{i=1}^\infty Tx_i \) is weak unconditional convergence if \(\sum \limits _{i=1}^\infty x_i\) is unconditional convergence. In this paper, we study \({\mathscr {A}}_{G}(E,F)\) , where G is an AL-space, by introducing two new operator related to Levi property, which are named Levi operator and \(\sigma \) -Levi operator respectively. Suppose \(T:E\rightarrow F\ge 0\) . We shows that if E has weak Fatou property then T is \(\sigma \) -Levi if and only if \(T\in {\mathscr {A}}_{G}(E,F)\) for every separable AL-space G if and only if \(T\in {\mathscr {A}}_{L_1[0,2\pi ]}(E,F)\) . Some relationships of two class operators and other operator classes are given. PubDate: 2022-05-10
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Abstract: Abstract The paper is devoted to such sets of \(E'\) on which every b order bounded disjoint sequence \((x_{n})\subset E\) converges uniformly to zero. We characterize this class of sets (b almost order (L) sets), consequently, we obtain some new characterizations of b weakly compact operators and KB spaces. Moreover, we present a dual version of Dunford Pettis theorem concerning relatively weakly compact subsets of a KB space, and show that in a Banach lattice whose dual has order continuous norm, every b almost order (L) set is relatively weakly compact. Furthermore, we prove that we can replace disjoint sequences appearing in the definition of b almost order (L) sets with positive weakly null ones, and we claim that a subset \(A \subset E'\) is b almost order (L) set if, and only if, every b order bounded positive weakly null sequence converges uniformly to zero on the subset A. PubDate: 2022-05-10
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Abstract: Abstract Given \(E_0, E_1, F_0, F_1, E\) rearrangement invariant function spaces, \(a_0\) , \(a_1\) , \(\mathrm {b}_0\) , \(\mathrm {b}_1\) , \(\mathrm {b}\) slowly varying functions and \(0< \theta _0<\theta _1<1\) , we characterize the interpolation spaces $$\begin{aligned} \big ({\overline{X}}^{{\mathcal {R}}}_{\theta _0,\mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{{\mathcal {R}}}_{\theta _1, \mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E},\quad \big ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{\mathcal L}_{\theta _1,\mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E} \end{aligned}$$ and $$\begin{aligned} \big ({\overline{X}}^{{\mathcal {R}}}_{\theta _0,\mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{{\mathcal {L}}}_{\theta _1, \mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E},\quad \big ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{\mathcal R}_{\theta _1,\mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E}, \end{aligned}$$ for all possible values of \(\theta \in [0,1]\) . Applications to interpolation identities for grand and small Lebesgue spaces, Gamma spaces and A and B-type spaces are given. PubDate: 2022-04-29
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Abstract: Abstract In this paper, we prove that Hadamard power of a even-order positive tensor has only one positive eigenvalue. We define a conditionally negative definite tensor and an m-invertible tensor. We prove that if \({\mathscr {A}} =(a_{i_1i_2\ldots i_m})\) is a conditionally negative definite and positive tensor, then \({\mathscr {A}}\) has exactly one positive eigenvalue. Next, we prove that if \({\mathscr {A}} =(a_{i_1i_2\ldots i_m})\) is an orthogonally decomposable tensor whose all entries are positive and has only one positive eigenvalue, then \({\mathscr {A}}^{\circ r} = (a^r_{i_1i_2\ldots i_m})\) also has only one positive eigenvalue for all \(r\in [0,1)\) . Also, the Hadamard inverse of \({\mathscr {A}}\) (with \(a_{i_1\dots i_m}\ne 0\) for all \(1\le i_1,\ldots , i_m\le n\) ) denoted by \({{\mathscr {A}}^\circ }^{(-1)}=(1/a_{i_1\dots i_m})\) is positive semi-definite. PubDate: 2022-04-29
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Abstract: Abstract Several recent papers investigated unbounded convergences in Banach lattices. In this paper, we first introduce un L- and M-weakly compact operators on Banach lattices, which contrast with L-(M-)weakly compact operators, by unbounded norm convergence. Then we research the relationship between these operators and L-(M-) weakly compact operators. Finally, we study the order and lattice structure of un L- and M-weakly compact operators. PubDate: 2022-04-25
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Abstract: Abstract Let \({\mathcal {S}}_{\alpha } (0 \le \alpha <\frac{\pi }{2} )\) stand for the set of all complex sector matrices and \(\sigma _1, \sigma _2\) be two operator means satisfying \(\sigma _1 \le \sigma _2.\) Except some other assertions, it is also shown that for \(A, B \in {\mathcal {S}}_{\alpha }, \) $$\begin{aligned} \Re (A\sigma _1 B)\le \sec ^2\alpha \ \Re (A\sigma _2 B) \end{aligned}$$ and $$\begin{aligned} \Re (A\sigma _2 B)^{-1}\le \sec ^2\alpha \ \Re (A\sigma _1 B)^{-1}. \end{aligned}$$ In addition, if \(\sigma _i^{*} \le \sigma _i,\) for \(i=1, 2\) and \(\Phi \) is a unital positive linear map, then $$\begin{aligned} \Phi \Re (A \sigma _1 B)^{-1}\le \sec ^2\alpha \ \Re \big (\Phi (A^{-1})\sigma _2\Phi (B^{-1})\big ). \end{aligned}$$ PubDate: 2022-04-25
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Abstract: Abstract In 1991 Soardi introduced a sequence of positive linear operators \(\beta _{n}\) associating to each function \(f\in C\left[ 0,1\right] \) a polynomial function which is closely related to the Bernstein polynomials on \(\left[ -1,+1\right] \) . One of the authors already studied the operators \(\beta _{n}\) in several papers. This paper is devoted to other properties of Soardi’s operators. We introduce a version \({\tilde{\beta }}_{n}\) which can be expressed in terms of the classical Bernstein operators and present the relations between \(\beta _{n}\) and \({\tilde{\beta }}_{n}\) . We derive Voronovskaja-type results for both \(\beta _{n}\) and \({\tilde{\beta }}_{n}\) . Furthermore, rates of convergence for \({\tilde{\beta }}_{n}\) , respectively \(\beta _{n}\) , are estimated. Finally, we study the first and second moments of \(\beta _{n}\) . PubDate: 2022-04-25
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Abstract: Abstract Based on the concept of unbounded absolute weak convergence, we give new characterizations of L-weakly compact sets. As applications, we find new properties of some classes of operators. Also, new characterizations of order continuous Banach lattices are obtained. PubDate: 2022-04-23
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Abstract: Abstract We show an asymptotic relation between the fractional Peetre type K-functional and the modulus of smoothness on a compact two-point homogeneous space. This extends both: the well known results on the spherical setting and results for the modulus of smoothness with positive integer degree of smoothness. This is a natural continuation of the research including a previous characterization, obtained by first author in 2019 (Carrijo in Positivity 24:761–777, 2020). An application of the asymptotic relation obtained is employed to prove a reciprocal result for the standard decrease property of the degree of the fractional modulus of smoothness. PubDate: 2022-04-22
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Abstract: Abstract We show continuity in generalized weighted Orlicz-Morrey spaces \(M^{\Phi ,\varphi }_{w}\) of sublinear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators with discontinuous data. PubDate: 2022-04-20
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Abstract: Abstract The class of weakly compact operators does not contain that of almost L-weakly compact operators. In this paper, we provide a complete answer by giving necessary and sufficient conditions for which every positive almost L-weakly compact operator \(T:E\rightarrow F\) between two Banach lattices is weakly compact. On the other hand, we investigate conditions under which the adjoint operator of every positive almost L-weakly compact operator is almost M-weakly compact. PubDate: 2022-04-19
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Abstract: Abstract Using a generalized Bochner type theorem for Olshanski spherical pairs, we prove a Lévy-Khinchin formula for the infinite dimensional Heisenberg group \(H_\infty = M(\infty , {\mathbb {C}})\times {\mathbb {R}}\) relatively to the action of the product group \(K_\infty =U(\infty )\times U(\infty )\) . PubDate: 2022-04-16
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Abstract: Abstract Several properties of subspaces \(J \subset X\) in a pair (X, J) with quotient lifting properties of different types are derived, giving emphasis to order-preserving properties for order unit spaces and spaces of affine continuous functions on Choquet simplexes. PubDate: 2022-04-10
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Abstract: Abstract Given two subsets X and Y of the real line with at least two points, BV(X) and BV(Y) denote the Banach spaces of all functions of bounded variation on X and Y, respectively. In this paper we study the 2-topological reflexivity of sets of (not necessarily linear) surjective isometries from BV(X) onto BV(Y). In particular, we obtain generalizations of the known results concerning 2-algebraical reflexivity of sets of surjective complex-linear isometries of BV(X)-spaces. PubDate: 2022-03-24
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Abstract: Abstract We establish the Schwarz inequality for a class of Banach *-algebras, and use it to derive some consequences, for example when a linear map between Banach *-algebras is a Jordan homomorphism. This applies to the class of Banach *-algebras \(\ell ^1(G,A;\alpha )\) arising from C*-dynamical systems \((A,G,\alpha )\) with \(\alpha \) an action of a discrete group G on a separable C*-algebra A. PubDate: 2022-03-19
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Abstract: Abstract This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from Albiac and Leránoz (J Math Anal Appl 374(2):394–401, 2011. https://doi.org/10.1016/j.jmaa.2010.09.048) we show that if \(X\) is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum \(\ell _{1}(X)\) has a unique unconditional basis up to a permutation, even without knowing whether \(X\) has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work. PubDate: 2022-03-19
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Abstract: Abstract We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such a topology is strictly finer than the coarse wedge topology and it turns every chain complete, rooted tree into a Fréchet–Urysohn, countably compact topological space. We show the rôle of such topology in the theory of weakly Corson and weakly Valdivia compacta. In particular, we give the first example of a compact space T whose every closed subspace is weakly Valdivia, yet T is not weakly Corson. This answers a question due to Ondřej Kalenda. PubDate: 2022-03-15
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Abstract: Abstract We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space E. The monotone convergence theorem, Fatou’s lemma, and the dominated convergence theorem are established; the analogues of the classical \({\mathscr {L}}^1\) - and \({\mathrm L}^1\) -spaces are investigated. The results extend earlier work by Wright and specialise to those for the Lebesgue integral when E equals the real numbers. The hypothesis on E that is needed for the definition of the integral and for the monotone convergence theorem to hold ( \(\sigma \) -monotone completeness) is a rather mild one. It is satisfied, for example, by the space of regular operators between a directed partially ordered vector space and a \(\sigma \) -monotone complete partially ordered vector space, and by every JBW-algebra. Fatou’s lemma and the dominated convergence theorem hold for every \(\sigma \) -Dedekind complete space. When E consists of the regular operators on a Banach lattice with an order continuous norm, or when it consists of the self-adjoint elements of a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space, then the finite measures as in the current paper are precisely the strongly \(\sigma \) -additive positive operator-valued measures. When E is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in our sense, but not conversely. Even when a measure falls into both categories, the domain of the integral as defined in this paper can properly contain that of any reasonably defined integral with respect to the vector measure using Banach space methods. PubDate: 2022-03-14
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Abstract: Abstract Uniform Kadec-Klee property, which takes an indispensable part in the researches of some mathematics branches, has attracted increasing extensive exploration and discussion. In this paper, necessary and sufficient conditions for uniform Kadec-Klee property in Orlicz-Lorentz sequence space equipped with Orlicz norm are given. PubDate: 2022-03-12
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Abstract: Abstract In this paper, we study (uniformly) mean ergodic composition operators on \(H^\infty (\mathbb {B}_n)\) . Under some additional assumptions, it is shown that mean ergodic operators have norm convergent iterates in \(H^\infty (\mathbb {B}_n)\) , and that they are always uniformly mean ergodic. PubDate: 2022-03-11
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