On generalized resolvents and \(Q\)-functions of symmetric linear relations (subspaces) in Hilbert space. (English) Zbl 0335.47014

Langer, Heinz K.; Textorius, B.

Pac. J. Math. 72, No. 1, 135-165 (1977).

Let \(H\) be a Hilbert space and let \(S\) be a symmetric linear relation in \(H\) [R. Arens, Pac. J. Math. 11, 9–23 (1961; Zbl 0102.10201)]. The notion of a \(Q\)-function of a symmetric operator in \(H\) with equal defect numbers was introduced by M. G. Kreĭn [Dokl. Akad. Nauk SSSR, n. Ser. 52, 651–654 (1946; Zbl 0063.03358)] and Sh.N. Saakyan [Dokl. Akad. Nauk Arm. SSR 41, 193–198 (1965; Zbl 0163.37804)]. In this paper the \(Q\)-function of a symmetric linear relation \(S\) is introduced and M. G. Kreĭn’s formula for the generalized resolvents of a symmetric linear operator is extended to the generalized resolvents of \(S\). In terms of the \(Q\)-function a necessary and sufficient condition for a minimal self-adjoint linear relation extension of \(S\) to be an operator follows. We also give a description, equivalent to a recent result of M. G. Kreĭn and I. E. Ovcharenko [Dopov. Akad. Nauk Ukr. RSR, Ser. A 1976, 881–884 (1976; Zbl 0334.47019)] of all generalized self-adjoint contraction resolvents of a symmetric non-densely defined contraction \(T\). Moreover generalized resolvents of \(T\) corresponding to minimal self-adjoint extensions with spectra in an interval \([\alpha,\beta]\), \(-\infty\leq \alpha\leq -1\), \(1\leq \beta\leq \infty\) are considered. As an application we prove some results about nonnegative linear relations, which completely correspond to results of E. A. Coddington [“Extension theory of formally normal and symmetric subspaces”, Mem. Am. Math. Soc. 134, 80 p. (1973; Zbl 0265.47023)].

Reviewer: Heinz K. Langer (Dresden)

MSC:

47B25	Linear symmetric and selfadjoint operators (unbounded)
47A10	Spectrum, resolvent
47A20	Dilations, extensions, compressions of linear operators

Citations:

Zbl 0102.10201; Zbl 0063.03358; Zbl 0163.37804; Zbl 0265.47023; Zbl 0334.47019