Generalized resolvents of quasi-self-adjoint contractive extensions of a Hermitian contraction (Q795295)

Let A be a Hermitian contraction in a Hilbert space H. \(T\in {\mathcal L}(\tilde H)\) (\(\widetilde H\supset H)\) is a quasi-selfadjoint contractive (qsc) extension of A if \(T\supset A\), \(T^*\supset A\) and \(\| T\| \leq 1\). Generalized resolvents \(R_{\lambda}=P(T-\lambda T)^{-1}|_ H\) of T, where P is an orthogonal projection onto H, are considered. The paper establishes a one-to-one correspondence between the family of all generalized resolvents of qsc-extensions of A and a certain class of holomorphic operator functions. In the case of selfadjoint contractive extensions of A the description of the family of generalized resolvents was given by \textit{M.G. Kreijn}, \textit{I. E. Ovcharenko} [Sib. Math. Zh. 18, 1032-1056 (1977; Zbl 0384.47006) and Dokl. Akad. Nauk SSSR 231, 1063- 1066 (1976; Zbl 0362.47009)].