Homogeneous multiplication operators on bounded symmetric domains.
The purpose of the authors is to study the spectral property of a family of multiplication tuples on the holomorphic function spaces with the reproducing kernel \(1-\langle z,w\rangle)^{-v}\) on the unit ball \(B^d\) of \(\mathbb{C}^d\), for \(0\leq v< \infty\). They also generalize this problem: Let \(D\) be an irreducible bounded symmetric domain of dimension \(d\) and let \(H_v(D)\) be the analytic continuation of the weighted Bergman spaces of holomorphic functions on \(D\). Consider the \(d\)-tuple \(M=(M_1,\dots, M_d)\) of multiplication operators by coordinate functions. The authors study the similar problem of spectral property of the \(d\)-tuple \(M\) on \(H_v(D)\). They find those parameters \(v\) for which the tuple \(M\) is subnormal and answer some open questions of \textit{B. Bagchi} and \textit{G. Misra} [ibid. 136, 171--213 (1996; Zbl 0867.47022)]. In particular, they prove that when \(D= B^d\), then \(B^d\) is a \(k\)-spectral set of \(M\) if and only if \(H_v(B^d)\) is the Hardy space or a weighted Bergman space. Also, in this case, the contractivity, hyponormality and subnormality of the multiplication tuples are investigated.
scientific article; zbMATH DE number 1974790
scientific article