Q-functions of Quasi-selfadjoint Contractions

Abstract

A bounded everywhere defined operator T in a Hilbert space ℌ is said to be a quasi-selfadjoint contraction or (for short) a qsc-operator, if T is a contraction and ker (T − T*) ≠ {0}. For a closed linear subspace $\mathfrak{N}$ of ℌ containing ran (T − T*) the operator-valued function Q_T (z)=$P_\mathfrak{A}$ (T − zI)⁻¹↾$\mathfrak{N}$ , |z|>1, where P$P_\mathfrak{A}$ is the orthogonal projector from ℌ onto $\mathfrak{N}$ , is said to be a Q-function of T acting on the subspace $\mathfrak{N}$ . The main properties of such Q-functions are studied, in particular the underlying operator-theoretical aspects are considered by using some block representations of the contraction T and analytical characterizations for such functions Q_T(z) are established. Also a reproducing kernel space model for Q_T(z) is constructed. In the special case where T is selfadjoint Q_T(z) coincides with the Q-function of the symmetric operator A ≔ T↾ (ℌ ⊖ $\mathfrak{N}$ ) and its selfadjoint extension T = T* in the usual sense.

To Heinz Langer on the occasion of his retirement

This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 203227, 208057) and the Dutch Organization for Scientific Research NWO (B 61-553).