# Spectral enclosures for a class of block operator matrices

## TL;DR

This paper develops new spectral enclosures for certain block operator matrices, including a Gershgorin-like theorem, with applications to J-frame operators, advancing understanding of their spectral properties.

## Contribution

It introduces novel spectral enclosures for block operator matrices with self-adjoint diagonal entries, extending Gershgorin's theorem to this operator class.

## Key findings

- Derived spectral enclosures for non-real spectra.
- Extended Gershgorin's circle theorem to block operator matrices.
- Applied results to analyze J-frame operators.

## Abstract

We prove new spectral enclosures for the non-real spectrum of a class of $2\times2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^*$ as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to $J$-frame operators.