Special operator classes and their properties

Abstract

We introduce some special operator classes and study in terms of Berezin symbols their properties. In particular, we give some characterizations of compact operators and Schatten-von Neumann class operators in terms of Berezin symbols. We also consider some classes of compact operators on a Hilbert space $H,$ which are generalizations of the well known Schatten-von Neumann classes of compact operators. Namely, for any number $p \in (0,\infty)$ and the sequence $w:=(w_{n})_{n\geq0}$ of complex numbers $w_{n},$ $n\geq 0,$ we define the following classes of compact operators on $H $: $$S_{p}^{w}(H)=\left\{ K\in S_{\infty}(H):\sum_{n=0}^{\infty}(s_{n} (K))^{p}w_{n}^{p}\hbox{ is convergent series }\right\}, $$ where $s_{n}(K)$ denotes the $n$th singular number of the operator $K$. The characterizations of these classes are given in terms of Berezin symbols.