Matrix multiplication operators on Banach function spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators $\Gamma$ on a Hilbert space. Our assumption on $\Gamma$ is expressed in terms of $\alpha$-boundedness for a Euclidean structure $\alpha$ on the underlying Banach space $X$. This notion is originally motivated by $\mathcal{R}$- or $\gamma$-boundedness of sets of operators, but, for example, any operator ideal from the Eucli...

Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued $L^1$-spaces into $L^\infty$-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on $L^p$-spaces o...

In this paper, we characterize the (left) multiplier algebra of a semidirect product algebra ${\mathcal A}={\mathcal B}\oplus {\mathcal I}$, where ${\mathcal I}$ and ${\mathcal B}$ are closed two-sided ideal and closed subalgebra of ${\mathcal A}$, respectively. As an application of this result we investigate the BSE-property of this class of Banach algebras. We then for two commutative semisimple Banach algebras ${\mathcal A}$ and ${\mathcal B}$ characterize the BSE-function...

In the first part of this paper we provide a self-contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of $C_0$-semigroups.

In this note we answer positively to two conjectures proposed by Nieraeth (2023) about the maximal operator on rescaled Banach function spaces. We also obtain a new criterion saying when the maximal operator bounded on a Banach function space $X$ is also bounded on the associate space $X'$.

In this work we investigate semigroups of operators acting on noncommutative $L^p$-spaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and $H^\infty$ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.

We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the solution x(\cdot) can be represented as a limit of sequence F(t/n)^{n} as n\to\infty in the weak operator topology, where a function F:[0,\infty)\to L(X) satisfies F'(0)y=Ay, y\in D(A). As a consequence, we deduce necessary and sufficient condi...

We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all functional Banach spaces in which every bounded analytic function is a pointwise multiplier. Next, we characterize (in various ways) the weighted composition operators among the bounded operators on such spaces, thus generalizing some well-kn...

In this paper, regularity properties, Strichartz type estimates to solutions of multipo{\i}nt Cauchy problem for linear and nonlinear abstract wave equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Hilbert space H, in which by choosing H and A we can obtain numerous classis of nonlocal initial value problems for wave equations which occur in a wide variety of physical systems.