Restriction and Factorization for Isometric and Symmetric Operators in Almost Pontryagin Spaces
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 S 1 of S. Every compressed resolvent of S 1 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 S 1 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.
KeywordsAlmost Pontryagin space restriction, factorization isometry, symmetric operator selfadjoint extension compressed resolvent
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