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