# Toeplitz operators on the domain $\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\}$ with $\mathrm{U}(2)\times\mathbb{T}^2$-invariant symbols

## TL;DR

This paper studies Toeplitz operators with specific invariant symbols on a symmetric domain of 2x2 matrices, showing they generate commutative algebras and deriving their spectra using representation theory.

## Contribution

It characterizes commutative Toeplitz operator algebras on a matrix domain with invariant symbols and provides spectral formulas via representation theory.

## Key findings

- Invariant symbols produce commutative Toeplitz algebras.
- Spectral formulas are derived using representation theory.
- Results apply to all weighted Bergman spaces on the domain.

## Abstract

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators.