L-resolvents of symmetric linear relations in Pontryagin spaces (2505.14883v1)

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.