Radial-like Toeplitz Operators on Cartan Domains of Type I

Abstract

Let \({\mathrm {D}}^{{\mathrm {I}}}_{n \times n}\) be the Cartan domain of type I which consists of the complex \(n \times n\) matrices Z that satisfy \(Z^*Z < I_n\). For a symbol \(a \in L^\infty ({\mathrm {D}}^{{\mathrm {I}}}_{n \times n})\) we consider three radial-like type conditions: 1) left (right) \({\mathrm {U}}(n)\)-invariant symbols, which can be defined by the condition \(a(Z) = a\big ((Z^*Z)^\frac{1}{2}\big )\) (\(a(Z) = a\big ((ZZ^*)^\frac{1}{2}\big )\), respectively), and 2) \({\mathrm {U}}(n) \times {\mathrm {U}}(n)\)-invariant symbols, which are defined by the condition \(a(A^{-1}ZB) = a(Z)\) for every \(A, B \in {\mathrm {U}}(n)\). We prove that, for \(n \ge 2\), 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 \(T_a\) and \(T_b\) commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for \(n \ge 2\), symbols a whose corresponding Toeplitz operators \(T_a\) are non-normal. We use these facts to prove the existence, for \(n \ge 2\), of commutative Banach non-\(C^*\) algebras generated by Toeplitz operators.