Radial Toeplitz operators on Bergman spaces

In this talk, we will first discuss Toeplitz operators acting on the Bergman space over

the unit disc D with radial symbols, which are defined as those satisfying a(z) = a(|z|)

for every z ∈ D. It was first discovered by Korenblum-Zhu that the Toeplitz operators

with such symbols mutually commute, and so generate a commutative C∗-algebra.

We will also discuss a higher dimensional generalization obtained on matrix domains

that generalize the unit disk. More precisely, we consider the Cartan domain of type I

DIn×n which consists of the n×n complex matrices Z satisfying Z∗Z < In. Bergman spaces

and Toeplitz operators will also be defined in this case, and we will consider symbols on

DIn×n that satisfy either of the following conditions

1. a(Z) = a((Z∗Z)12) for all Z ∈ DIn×n.

2. a(Z) = a((ZZ∗)12) for all Z ∈ DIn×n.

We will show that for n ≥ 2, these conditions are not equivalent. However, both are

equivalent to invariance with respect to a corresponding subgroup of the biholomorphisms

fixing the origin. We will use representation theory to obtain several algebras generated

by Toeplitz operators with these special kind of symbols. This will provide commutative

and non-commutative algebras that can be either C∗ or only Banach.