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H. de Snoo, H. Woracek:
"Restriction and factorization for isometric and symmetric operators in almost Pontryagin spaces";
in: "ASC Report 05/2015", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2015, ISBN: 978-3-902627-08-7, 1 - 43.

Abstract. We investigate symmetric linear relations in almost Pontryagin spaces. A notion of restriction and factorization is introduced. It applies to both spaces and relations.
The question under consideration is how symmetric extensions and inner
products involving resolvents ("compressed resolvents") behave when a
restriction-factorization process is applied. The main result, which holds under some natural conditions, is for a symmetric relation S and a restricted and factorized relation S1 of S. Every compressed resolvent of S1 can be realized as the compressed resolvent of a restriction-factorization of a symmetric extension
of the original relation S. However, in general not every symmetric
extension of S1 coincides with the restriction-factorization of some symmetric extension of S. The difficulties one encounters, as well as the methods employed to overcome them, are mainly of geometric nature and are specific for the indefinite and degenerated situation.
The present results form the core needed to understand minimality notions for symmetric and selfadjoint linear relations in almost Pontryagin spaces.

Almost Pontryagin space, restriction, factorization, isometry, symmetric operator, selfadjoint extension, compressed resolvent

http://www.asc.tuwien.ac.at/preprint/2015/asc05x2015.pdf