New York Journal of Mathematics
Volume 23 (2017) 555-582


Sergey M. Zagorodnyuk

Unitary extensions of pairs of commuting isometric operators and their generalized resolvents

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Published: May 21, 2017
Keywords: Isometric operator, unitary extension, generalized resolvent, spectral family
Subject: 47B15

In this paper we study a pair V1,V2 of commuting isometric operators in a Hilbert space H, which are not necessarily defined on the whole space. An old question: is there a possibility for an extension of V1,V2 to a pair of commuting unitary operators U1,U2 in a Hilbert space \widetilde H⊇ H? In the case of a unitary V2 we present a transparent criterion in terms of the original space H. The general case is discussed, as well. We introduce a notion of a generalized resolvent for V1,V2. Characteristic properties of the generalized resolvent in terms of H are obtained. In the case of a unitary V2 all generalized resolvents are parametrized.

Author information

School of Mathematics and Informatics, V.N. Karazin Kharkiv National University, Kharkiv 61022, Ukraine