Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho$ behaves like a polynomial in $\rho$, and consider the mapping properties between amalgams of kernel operators where the kernel $kerK(x,y)$ behaves like $d(x,y{)}^{-a}$ when $d(x,y)\le 1$ and like $d(x,y{)}^{-b}$ when $d(x,y)\ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

Let $L$, $M$ be Archimedean Riesz spaces and ${ℒ}_b(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of ${ℒ}_b(L,M)$ to be an ordered vector subspace $ℒ$ of ${ℒ}_b(L,M)$ such that the elements of $ℒ$ preserve disjointness and any pair of operators in $ℒ$ has a supremum in ${ℒ}_b(L,M)$ that belongs to $ℒ$. It turns out that the lattice operations in any Lamperti Riesz subspace $ℒ$ of ${ℒ}_b(L,M)$ are given pointwise, which leads to a generalization of the classic Radon-Nikod’ym theorem for Riesz homomorphisms....

Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, ${\mathscr{H}}_{Nb}\left(E\right)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on ${\mathscr{H}}_{Nb}\left(E\right)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $\phi \left(D\right){\equiv }^t\phi$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on ${\mathscr{H}}_{Nb}\left(E\right)$ is PDE-preserving for a set ℙ ⊆ Exp(F) if it...

The class of Rosenthal linear relations in normed spaces is introduced and studied in terms of their first and second conjugates. We investigate the relationship between a Rosenthal linear relation and its conjugate. In this paper, we also study the semi-Tauberian linear relations following the pattern followed for the study of the Tauberian linear relations. We prove that the semi-Tauberian linear relations share some of the properties of Tauberian linear relations and they are related to Rosenthal...

A scattered element of a Banach algebra is an element with at most countable spectrum. The set of all scattered elements is denoted by (). The scattered radical ${}_{sc}\left(\right)$ is the largest ideal consisting of scattered elements. We characterize in several ways central elements of modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) () + () ⊂ (); (ii) ()() ⊂ (); (iii) $[\left(\right),]{\subset }_{sc}\left(\right)$.

Let $\mu$ be a finite positive measure on the unit disk and let $j\ge 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{\mu }^{\left(j\right)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$ on the Bergman space $A^2$, and then give a sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^2$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D,\mathrm{d}A)$ and $1<p<\infty$, we define the generalized Toeplitz...

A full description of the membership in the Schatten ideal $S_p\left(A{²}_{\omega }\right)$ for 0 < p < ∞ of Toeplitz operators acting on large weighted Bergman spaces is obtained.