Multipliers for operator-valued Bessel sequences, generalized Hilbert-Schmidt and trace classes

TL;DR

This paper extends the theory of operator multipliers by incorporating operator-valued Bessel sequences and generalizing Hilbert-Schmidt and trace class operators, broadening the scope of operator analysis in Hilbert spaces.

Contribution

It introduces a new framework for multipliers involving operator-valued Bessel sequences and generalizes classical Hilbert-Schmidt and trace class operators.

Findings

Established conditions for convergence of the operator series.

Extended classical classes to include operator-valued sequences.

Provided new characterizations of generalized Hilbert-Schmidt and trace classes.

Abstract

Let $\{\lambda_n\}_n \in \ell^\infty(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}\lambda_n (x_n\otimes \bar{y_n})$, where $\{x_n\}_n$, $\{y_n\}_n$ are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \cite{BALAZS3} generalized this by replacing $\{x_n\}_n$ and $\{y_n\}_n$ by Bessel sequences. In this paper, we generalize this by studying the operators of the form $\sum_{n=1}^{\infty}\lambda_n (A^*_nx_n\otimes \bar{B^*_ny_n})$, where $\{A_n\}_n$ and $\{B_n\}_n$ are operator-valued Bessel sequences and $\{x_n\}_n$, $\{y_n\}_n$ are sequences in the Hilbert space such that $\{\|x_n\|\|y_n\|\}_n \in \ell^\infty(\mathbb{N})$. We next generalize the classes of Hilbert-Schmidt and trace class operators.