Résolvant généralisé et théorie spectrale. (Generalized resolvent and spectral theory). (French) Zbl 0694.47002

Mbekhta, Mostafa

J. Oper. Theory 21, No. 1, 69-105 (1989).

Let H be a Hilbert space and let A be a closed operator with domain D(A) and range R(A) in H. If U is a subset of \({\mathbb{C}}\), we say that A has a generalized resolvent operator \(R_ g(A,\lambda)\) in U if \(\forall \lambda \in U\), \(R_ g(A,\lambda)\) is a bounded operator of H into D(A) such that \[ (A-\lambda I)R_ g(A,\lambda)(A-\Lambda I)=A-\lambda I,\quad R_ g(A,\lambda)(A-\lambda I)R_ g(A,\lambda)=R_ g(A,\lambda). \] The regular (generalized resolvent) set of A is defined by \[ reg(A)=\{\lambda \in {\mathbb{C}}:\text{ A has a generalized resolvent, analytic on a neighborhood of }\lambda\}. \] The generalized spectrum of A is \(\sigma_ g(A)={\mathbb{C}}\setminus reg(A).\)
The author studies a generalized spectral theory in which the notion of inverse is replaced by that of relative inverse or generalized inverse first introduced by F. V.. Atkinson [Acta Sci. Math. szeged 15, 38- 56 (1953; Zbl 0052.125)]. Conditions under which reg(A) is equal to the resolvent set of A are given. The perturbations of \(\sigma_ g(A)\) by a quasinilpotent operator is discussed. Also a notion of spectral mapping theorem is proved for \(\sigma_ g(A)\). The distance of a point from \(\sigma_ g(A)\) is evaluated.

Reviewer: K.Seddighi

MSC:

47A10	Spectrum, resolvent
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A53	(Semi-) Fredholm operators; index theories

Keywords:

quasi-Fredholm operator; generalized resolvent operator; generalized spectrum; generalized spectral theory; relative inverse; generalized inverse; quasinilpotent operator; spectral mapping theorem

Citations:

Zbl 0052.125