Generalized resolvent matrices and spaces of analytic functions

Abstract

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2 × 2-matrix valued analytic functions which satisfy a certain kernel condition.

Resolvent Matrices in Degenerated Inner Product Spaces

“…With exception of the last statement all assertions are proved similar as the corresponding results in [KW3]. To prove the last assertion note that for any…”

“…Choose a fundamental symmetry J on P and define ( · , · ) := [J · , · ]. We use the following notation (compare [KW3]):…”

“…In [KW3] the element u was allowed to be a so -called generalized element, which leads to a natural characterization of those matrix functions W (z) which appear as resolvent matrices. For a particular subclass of the set of all resolvent matrices, namely for those W where u ∈ P, a related characterization can be found in [KL].…”

“…In Section 2 we provide the theory of generalized elements and triplet spaces for a closed symmetric relation with defect index (n, n), n ∈ N, in a Pontryagin space, which is similar to the considerations of [KW3] in the case of defect (1, 1). This notion is used to define triplet spaces for a degenerated inner product space P. Section 3 is concerned with the study of regularized resolvents of S ⊆ P 2 .…”

Unitary Boundary Pairs for Isometric Operators in Pontryagin Spaces and Generalized Coresolvents

“…An elegant proof of the formula for generalized resolvents of a nonstandard Pontryagin space symmetric operator with deficiency index (1,1) given by H. de Snoo was presented in [34]. In [5] a description of regular generalized resolvents of a nonstandard Pontryagin space isometric operator was given by the method of boundary triples.…”