Radial-like Toeplitz operators on Cartan domains of type I

Abstract

Let ${\mathrm{D}}_{n\times n}^{\mathrm{I}}$ be the Cartan domain of type I which consists of the complex $n\times n$ matrices $Z$ that satisfy $Z^{\ast }Z<I_n$. For a symbol $a\in L^{\infty }({\mathrm{D}}_{n\times n}^{\mathrm{I}})$ we consider three radial-like type conditions: 1) left (right) $\mathrm{U}(n)$-invariant symbols, which can be defined by the condition $a(Z)=a((Z^{\ast }Z{)}^{\frac{1}{2}})$ ($a(Z)=a((Z{Z}^{\ast }{)}^{\frac{1}{2}})$, 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-$C^{\ast }$ algebras generated by Toeplitz operators.