L-resolvents of symmetric linear relations in Pontryagin spaces

TL;DR

This paper extends the theory of L-resolvent matrices for symmetric linear relations to Pontryagin spaces, connecting boundary triples with Krein--Saakyan theory and applying results to canonical systems.

Contribution

It introduces new connections between boundary triples and L-resolvent matrices for relations in Pontryagin spaces, extending existing formulas.

Findings

Extended L-resolvent matrix formulas to Pontryagin spaces.

Connected boundary triple theory with Krein--Saakyan theory.

Applied results to minimal relations generated by canonical systems.

Abstract

Let A be a closed symmetric operator with the deficiency index (p,p), $p<\infty$, acting in a Hilbert space H and let L be a subspace of H. The set of L-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge L was described by Krein and Saakyan. The Krein--Saakyan theory of L-resolvent matrix was extended by Shmul'yan and Tsekanovskii to the case of improper gauge L and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Krein--Saakyan theory of L-resolvent matrices for symmetric linear relations with improper gauges in Pontryagin spaces. We extend the known formula for the L-resolvent matrix in terms of boundary operators to this class of relations. The results are applied to the minimal linear relation generated by a canonical system.