Banach Journal of Mathematical Analysis最新文献

Abstract

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

The Lipschitz-free space ({mathcal {F}}(M)) has an F.D.D. when M is a separable ({mathcal {L}}_1)-Banach space, or when (Msubset {mathbb {R}}^n) is a somewhat regular subset. The interplay between the existence of extension operators for Lipschitz maps and the ((pi ))-property in Lipschitz-free spaces is investigated. If M is an arbitrary metric space, then ({mathcal {F}}(M)) has the ((pi ))-property up to a universal logarithmic factor. It follows in particular that the ((pi ))-property up to a logarithmic factor fails to imply the approximation property. A list of commented open problems is provided.

We show that, for a class of locally solid topologies on vector lattices, a topologically convergent net has an embedded sequence that is unbounded order convergent to the same limit. Our result implies, and often improves, many of the known results in this vein in the literature. A study of metrisability and submetrisability of locally solid topologies on vector lattices is included.

Let A and B be two bounded linear operators on a Hilbert space. B is called the skew commutator of A if (_{*}[A, B]=AB-BA^{*}=0.) In this paper, we completely characterize when a Toeplitz operator on the Hardy space is a skew commutator of a Hankel operator and when a Hankel operator on the Hardy space is a skew commutator of a Toeplitz operator. Moreover, we also obtain a necessary and sufficient condition for the product of a Hankel operator and a Toeplitz operator to be self-adjoint on the Hardy space.

We characterize (vanishing) Fock–Carleson measures by products of functions in Fock spaces. We also study the boundedness of Berezin-type operators from a weighted Fock space to a Lebesgue space. Due to the special properties of Fock–Carleson measures, the boundedness of Berezin-type operators on Fock spaces is different from the corresponding results on Bergman spaces.

We establish the existence of a weak solution for a strongly coupled, nonlinear Stokes–Maxwell system, originally proposed by Nika and Vernescu (Z Angew Math Phys 71(1):1–19, 2020) in the three-dimensional setting. The model effectively couples the Stokes equation with the quasi-static Maxwell’s equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is premised on: (i) the augmented variational formulation of Maxwell’s equations, (ii) the definition of a new function space for the magnetic induction and the verification of a Poincar’e-type inequality, and (iii) the deployment of the Altman–Shinbrot fixed point theorem when the magnetic Reynolds number, ({text {R}_{text {m}}},) is small.

The concept of disjointness preserving mappings has proved to be a useful unifying idea in the study of Banach–Stone type theorems. In this paper, we examine disjointness preserving relations between sets of continuous functions (valued in general topological spaces). Under very mild assumptions, it is shown that a disjointness preserving relation is completely determined by a Boolean isomorphism between the Boolean algebras of regular open sets in the domain spaces. Building on this result, certain Banach–Stone type theorems are obtained for disjointness preserving relations. From these, we deduce a generalization of Kaplansky’s classical theorem concerning order isomorphisms to sets of continuous functions with values topological lattices. As another application, we prove some results on the characterization of nonvanishing preservers. Throughout, the domains of the function spaces need not be compact.

Abstract

This paper investigates the embedding relationships between Wiener amalgam spaces and classical spaces, including Sobolev spaces, local Hardy spaces, Besov spaces, and (alpha ) -modulation spaces. By establishing exact conditions, we provide a detailed characterization of the embeddings between Wiener amalgam spaces and these classical spaces, particularly the most general case when (alpha =0) , which extend the main results obtained by Guo–Wu–Yang–Zhao (J Funct Anal 273(1):404–443, 2017). Furthermore, we discuss the embedding relationship between Wiener amalgam spaces and Triebel–Lizorkin spaces (F_{p,r}^{s}) when (0<pleqslant 1) .

Abstract

In this paper, we focus on a class of fractional type integral operators that can be served as extensions of Riesz potential with kernels $$begin{aligned} K(x,y)=frac{Omega _1(x-A_1 y)}{|x-A_1 y |^{{n}/{q_1}}} cdots frac{Omega _m(x-A_m y)}{|x-A_m y |^{{n}/{q_m}}}, end{aligned}$$ where (alpha in [0,n)) , ( mgeqslant 1) , (sum limits _{i=1}^mfrac{n}{q_i}=n-alpha ) , ({A_i}^m_{i=1}) are invertible matrixes, (Omega _i) is homogeneous of degree 0 on (mathbb R^n) and (Omega _iin L^{p_i}(S^{n-1})) for some (p_iin [1,infty )) . Under appropriate assumptions, we obtain the weighted (L^p(mathbb R^n)-L^q(mathbb R^n)) estimates as well as weighted (H^p(mathbb R^n)-L^q(mathbb R^n)) estimates of the commutators for such operators with BMO-type function when (frac{1}{q}=frac{1}{p}-frac{alpha }{n}) . In addition, we acquire the boundedness of these operators and their commutators with a function in Campanato spaces on Orcliz–Morrey spaces as well as the compactness for such commutators in a special case: (m=1) and (A=I) .

In this article, the authors establish a non-smooth atomic decomposition of Triebel–Lizorkin-type spaces and, as a by-product, a non-smooth atomic decomposition of subspaces of BMO spaces is obtained. An application of this decomposition method to the boundedness of Marcinkiewicz integral operators is also presented.