Q-functions of Hermitian contractions of Krein-Ovcharenko type

Abstract

Operator valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space H. The definition is related to the investigations of M.G. Kreın and I.E. Ovcharenko on the so-called Qµ- and QM -functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Qµ- and QM -functions in the general operator valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function theoretic properties of the corresponding Q-functions.