Radial-like Toeplitz operators on Cartan domains of type I

Abstract

Let D I n×n be the Cartan domain of type I which consists of the complex n×n matrices Z that satisfy Z * Z < In. For a symbol a ∈ L ∞ (D I n×n ) we consider three radial-like type conditions: 1) left (right) U(n)-invariant symbols, which can be defined by the condition a, respectively), and 2) U(n) × U(n)-invariant symbols, which are defined by the condition a(A −1 ZB) = a(Z) for every A, B ∈ U(n). We prove that, for n ≥ 2, these yield different sets of symbols. If a satisfies 1), either left or right, and b satisfies 2), then we prove that the corresponding Toeplitz operators Ta and T b commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for n ≥ 2, symbols a whose corresponding Toeplitz operators Ta are non-normal. We use these facts to prove the existence, for n ≥ 2, of commutative Banach non-C * algebras generated by Toeplitz operators.