Q-functions of Quasi-selfadjoint Contractions

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 N of H containing ran (T − T*) the operator-valued function QT(z) = PN(T − zI)^−1↾N, |z| > 1, where PN is the orthogonal projector from H onto N, is said to be a Q-function of T acting on the subspace 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 QT(z) are established. Also a reproducing kernel space model for QT(z) is constructed. In the special case where T is selfadjoint QT(z) coincides with the Q-function of the symmetric operator A := T↾(H ⊖ N) and its selfadjoint extension T = T* in the usual sense.