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

Abstract

Let $D$ be the irreducible bounded symmetric domain of $2\times 2$ complex matrices that satisfy $Z{Z}^{\ast }<I_2$. 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 $C^{\ast }$-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.