Some problems in operator theory on bounded symmetric domains
Let \(T\) be a commuting \(d\)-tuple of bounded and linear operators on a separable Hilbert space and \({\mathcal H}\) a Hilbert space of holomorphic functions (on a domain in \({\mathbb C}^d\)), invariant under the \(d\)-tuple \(Z\) of operators on \({\mathcal H}\) of multiplications by the coordinates and having a reproducing kernel \(K(x,\bar y)\). It is the aim of the paper under review to discuss conditions in order to assure the existence of the following operator model: \(T^*\) is unitarily equivalent to the restriction of the backward multishift \((Z^*)^\infty\) of infinite multiplicity to an invariant subspace. The problem was completely solved by \textit{C. Ambrozie}, \textit{M. Engliš} and \textit{V. Müller} [J. Oper. Theory 47, 287--302 (2002; Zbl 1019.47015)] provided that \(1/K(x,\bar y)\) is a polynomial in \(x,\bar y\). Since this last condition seems to be too restrictive, another approach, developed by \textit{J. Arazy} and \textit{M. Engliš} [Trans. Am. Math. Soc. 355, 837--864 (2003; Zbl 1060.47013)], provides a solution if \({\mathcal H}\) admits a \({1\over k}\)-calculus. On bounded symmetric domains, in view of the Faraut-Koranyi formula related to the reproducing kernel \(K_m\) of the Peter-Weyl space with signature \(m\), a \({1\over k}\)-calculus is obtained by a substantial strengthening of a result in the above mentioned paper. A possible generalization to bounded symmetric domains of the theory of row contractions developed by \textit{W. Arveson} [J. Reine Angew. Math. 522, 173--236 (2000; Zbl 0951.47008)] is finally considered.
scientific article; zbMATH DE number 2093654