In this article, we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents $p$ and $q$, which depend on the type $p$ and cotype $q$ of the underlying Banach spaces. In a previous article, we considered $L^p$-$L^q$ multiplier theorems. In the current article, we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the $L^p$-$L^q$ setting as well.

We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties, then the boundedness of the multiplier operator extrapolates to other values of $p$ and $q$ for which $\frac{1}{p}-\frac{1}{q}$ remains constant.

For an arbitrary nonaffine operator convex function defined on the nonnegative real line and satisfying $f\left(0\right)=0$, we characterize the bijective maps on the set of all positive definite operators preserving a new version of quantum $f$-divergence. We also determine the structure of all transformations leaving this quantity invariant on quantum states for any strictly convex functions with the properties $f\left(0\right)=0$ and ${lim\hspace{0.17em}}_{x\to \infty }f\left(x\right)/x=\infty$. Finally, we derive the corresponding result concerning those transformations on the set of positive semidefinite operators. We emphasize that all the results are obtained for finite-dimensional Hilbert spaces.

In this article, we study dynamical and $C^{\ast }$-algebraic properties of self-similar group actions on finite directed graphs. We investigate the structure of limit dynamical systems induced from group actions on graphs, and we deduce conditions of group actions and graphs for the groupoid $C^{\ast }$-algebras defined by limit dynamical systems to be simple, separable, purely infinite, nuclear, and satisfying the universal coefficient theorem.

Using Gaussian processes, we define a very general convolution product of functionals on Wiener space and we investigate fundamental relationships between the generalized Fourier–Feynman transforms and the generalized convolution products. Using two rotation theorems of Gaussian processes, we establish that both of the generalized Fourier–Feynman transform of the generalized convolution product and the generalized convolution product of the generalized Fourier–Feynman transforms of functionals on Wiener space are represented as products of the generalized Fourier–Feynman transforms of each functional, with examples.

This article deals with trigonometric functional equations on hypergroups. We describe the general continuous solution of sine and cosine addition formulas and a so-called sine-cosine functional equation on a locally compact hypergroup in terms of exponential functions, sine functions, and second-order generalized moment functions.

We study some properties of Lipschitz mappings which admit factorization through an operator ideal. Lipschitz cross norms have been established from known tensor norms in order to represent certain classes of Lipschitz mappings. Inspired by the definition of $p$-summing linear operators, we derive a new class of Lipschitz mappings that is called strictly Lipschitz $p$-summing.

In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, $\left(T_q^p\right({\mathbb{R}}_{+}^{n+1}){)}^{*}=T_{q\text{'}}^{p\text{'}}({\mathbb{R}}_{+}^{n+1})$ for $1<p,q<\infty$, and $\left(T_{\infty }^1\right({\mathbb{R}}_{+}^{n+1}){)}^{*}=\mathcal{C}({\mathbb{R}}_{+}^{n+1})$, $\left(T_q^1\right({\mathbb{R}}_{+}^{n+1}){)}^{*}=T_{q\text{'}}^{\infty }({\mathbb{R}}_{+}^{n+1})$ for $1<q<\infty$, where $\mathcal{C}\left({\mathbb{R}}_{+}^{n+1}\right)$ denotes the Carleson measure space on ${\mathbb{R}}_{+}^{n+1}$. We prove that $\left({\mathcal{C}}_v\right({\mathbb{R}}_{+}^{n+1}){)}^{*}=T_{\infty }^1({\mathbb{R}}_{+}^{n+1})$, which we introduced recently, where ${\mathcal{C}}_v\left({\mathbb{R}}_{+}^{n+1}\right)$ is the vanishing Carleson measure space on ${\mathbb{R}}_{+}^{n+1}$. We also give the characterizations of $T_q^{\infty }\left({\mathbb{R}}_{+}^{n+1}\right)$ by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on $L^q\left({\mathbb{R}}^n\right)$ of the paraproduct ${\pi }_F$ associated with the tent space $T_q^{\infty }\left({\mathbb{R}}_{+}^{n+1}\right)$, and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces $T_2^p\left({\mathbb{R}}_{+}^{n+1}\right)$ $(1\le p\le \infty )$ and $L^p\left({\mathbb{R}}^n\right)$, $H^p\left({\mathbb{R}}^n\right)$ and $\mathit{BMO}\left({\mathbb{R}}^n\right)$ spaces.

Let $A$ be a Banach algebra, and let $\mathcal{X}$ be a left Banach $A$-module. In this paper, using the notation of point multipliers on left Banach modules, we introduce a certain type of spectrum for the elements of $\mathcal{X}$ and we also introduce a certain subset of $\mathcal{X}$ which behaves as the set of invertible elements of a commutative unital Banach algebra. Among other things, we use these sets to give some Gleason–Kahane–Żelazko theorems for left Banach $A$-modules.

We introduce the concept of Riesz-dual sequences for g-frames. In this paper we show that, for any sequence of operators, we can construct a corresponding sequence of operators with a kind of duality relation between them. This construction is used to prove duality principles in g-frame theory, which can be regarded as general versions of several well-known duality principles for frames. We also derive a simple characterization of a g-Riesz basic sequence as a g-R-dual sequence of a g-frame in the tight case.

The aim of this paper is to investigate asymmetric truncated Toeplitz operators with $L^2$-symbols between two different model spaces given by inner functions such that one divides the other. The class of symbols corresponding to the zero operator is described. Asymmetric truncated Toeplitz operators are characterized in terms of operators of rank at most $2$, and the relations with the corresponding symbols are studied.

We generalize and study, within the framework of quantum mechanics and working with $1$-dimensional, manifestly non-Hermitian Hamiltonians $H=-d^2/d{x}^2+V$, the traditional class of exactly solvable models with local point interactions $V=V\left(x\right)$. We discuss the consequences of the use of nonlocal point interactions such that $\left(Vf\right)\left(x\right)=\int K(x,s)f\left(s\right)ds$ by means of the suitably adapted formalism of boundary triplets.

Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^2\left(G\right)$ have linearly independent left translations. In particular, we establish some results for the case when $G$ has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.