${\sL}$-resolvents and pseudo-spectral functions of symmetric linear relations in Hilbert spaces

Abstract

Let $A$ be a closed symmetric operator with the deficiency index $(p,p)$, $p<\infty$, acting in a Hilbert space $\sH$ and let $\sL$ be a subspace of $\sH$. The set of $\sL$-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge $\sL(\subset\sH)$ was described by Kre\u{\i}n and Saakyan. The Kre\u{\i}n--Saakyan theory of $\sL$-resolvent matrices was extended by Shmul'yan and Tsekanovskii to the case of improper gauge $\sL(\not\subset\sH)$ 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 Kre\u{\i}n--Saakyan theory of $\sL$-resolvent matrices for symmetric linear relations with improper gauges in Hilbert spaces and extend the known formula for the $\sL$-resolvent matrix in terms of boundary operators to this class of relations. Descriptions of spectral and pseudo-spectral functions of symmetric linear relations with improper gauges are given. The results are applied to linear relations generated by a canonical system.