# Trace class perturbation of closed linear relations

## TL;DR

This paper investigates trace class perturbations of closed linear relations in Hilbert spaces, introducing new characterizations and criteria for such perturbations using orthogonal projections and block operator matrices.

## Contribution

It introduces the concept of trace class perturbation for closed relations and provides new equivalent characterizations using block operator matrices.

## Key findings

- Characterizations of compact and trace class block operator matrices.
- Equivalent criteria for trace class perturbation of closed linear relations.
- Use of orthogonal projections to define trace class perturbations.

## Abstract

This paper studies trace class perturbation of closed linear relations in Hilbert spaces. The concept of trace class perturbation of closed relations is introduced by orthogonal projections. Equivalent characterizations of compact and trace class block operator matrices are first given in terms of their elements, separately. By using them, several equivalent and sufficient characterizations of trace class perturbation of closed linear relations are obtained.