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.
