Radial-like Toeplitz operators on Cartan domains of type I (Q6610410)

For \(n\in\mathbb{N}\), let \(D_{n\times n}^{\mathrm I}\) be the Cartan domain of type I, that is the domain of all complex \(n\times n\) matrices \(Z\) satisfying \(Z^*Z<I_n\) (equivalently, \(ZZ^*<I_n\)). It is known that \(D_{n\times n}^{\mathrm I}\) is an irreducible circled bounded symmetric domain and that \(D_{1\times 1}^{\mathrm I}\) coincides with the unit disk \(\mathbb{D}\) in the complex plane. The paper is devoted to the study of Toeplitz operators with symbols \(a\in L^\infty(D_{n\times n}^{\mathrm I})\) on Bergman spaces. Note that a symbol \(a\in L^\infty(\mathbb{D})\) is radial if and only if \(a(tz)=a(z)\) for all \(t\in\mathbb{T}=\partial\mathbb{D}\) and \(z\in\mathbb{D}\). \N\NThe authors considers three natural generalizations of radial symbols for \(n>1\): \N\begin{itemize}\N\item[1)] the symbols satisfying \(a(Z)=a((Z^*Z)^{1/2})\) for \(Z\in D_{n\times n}^{\mathrm I}\); \N\item[2)] the symbols satisfying \(a(Z)=a((ZZ^*)^{1/2})\) for \(Z\in D_{n\times n}^{\mathrm I}\); \N\item[3)] the symbols satisfying \(a(A^{-1}ZB)=a(Z)\) for all \(A,B\in U(n)\) and \(Z\in D_{n\times n}^{\mathrm I}\). \N\end{itemize}\NFor \(n=1\) all three classes coincide with the class of radial symbols. However, it is shown in the paper under review that the above three classes are pairwise distinct. Further, it is shown that if \(a\) belongs to the first or the second class and \(b\) belongs to the third class, then the corresponding Toeplitz operators \(T_a\) and \(T_b\) commute on every weighted Bergman space. Among all symbols belonging either to the first or to the second class, there exists symbols \(a\) such that \(T_a\) are non-normal. These facts are used to prove the existence of commutative Banach non-\(C^*\)-algebras generated by Toeplitz operators on Bergman spaces.\N\NFor the entire collection see [Zbl 1531.47001].