Homogeneous tuples of multiplication operators on twisted Bergman spaces
Let \(\Omega_{n,m}\) be the domain of all \(n\times m\) contractive complex matrices \((m\geq n\geq 1)\), and let \(B\) be the Bergman kernel on this domain. Let also \({\mathcal W}\) be the set of those \(\lambda\geq 0\) for which \(B^{\lambda/(m+n)}\) is the reproducing kernel of a functional space \({\mathcal H}^{(\lambda)}\) (which is called a twisted Bergman space). In this paper the authors study the \(mn\)-tuple \(M^{(\lambda)}\) of operators on \({\mathcal H}^{(\lambda)}\) whose components are the multiplications by the \(mn\)-coordinate functions. They are concerned with the problem of determining the set of those \(\lambda\in{\mathcal W}\) for which \(M^{(\lambda)}\) is bounded, and moreover, subnormal. Other problems, as for instance when the Taylor spectrum of \(M^{(\lambda)}\) is a \(k\)-spectral set, are also examined. Let us mention that most of the techniques developed in this paper apply to the larger class of bounded Cartan domains.
scientific article; zbMATH DE number 871742