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Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux. (Generalization of the Kato decomposition to paranormal and spectral operators). (French) Zbl 0657.47038
In the well-known paper [J. Analyse Math. 6, 261-322 (1958; Zbl 0090.090)] T. Kato proved that if A is a semi-Fredholm operator with domain D(A) and range R(A) in a separable Hilbert space H then there exists a decomposition $$H=M\oplus N$$ where (a) M,N are invariant under A, (b) $$A| M$$ is regular (i.e. R(A$$| M)$$ is closed and contains the null spaces of all the iterates of $$A| M)$$, (c) $$N\subseteq D(A)$$ and $$A| N$$ is nilpotent of degree d. Such a decomposition is known as a Kato decomposition of degree d. Operators which admit such a decomposition were characterised in [J. P. Labrousse, Rend. Circ. Mat. Palermo, II. Ser. 29, 161-258 (1980; Zbl 0474.47008)] and are called quasi-Fredholm of degree d. In the present paper the author discusses some variations of Kato’s Theorem: for instance, he proves that if A is a spectral operator in the sense of Dunford, there is a Kato decomposition with the conditions (b) and (c) replaced by b’) $$A| M$$ is invertible, c’) $$A| N$$ is quasi-nilpotent.
Reviewer: W.D.Evans

##### MSC:
 47B40 Spectral operators, decomposable operators, well-bounded operators, etc. 47B20 Subnormal operators, hyponormal operators, etc. 47A53 (Semi-) Fredholm operators; index theories
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##### References:
 [1] Colojoara, Theory of generalized spectral operators (1968) [2] Apostol, Rev. Roumaine Math. Pures Appl. 21 pp 813– (1976) [3] Albrecht, Glasgow Math. J. 23 pp 91– (1982) [4] Vrbova, Czechoslovak Math. J. 23 pp 483– (1973) [5] Vasilescu, Analytic functional calculus and spectral decompositions (1982) · Zbl 0495.47013 [6] Nashed, Generalized inverses and applications pp 325– (1976) · doi:10.1016/B978-0-12-514250-2.50013-5 [7] Dunford, Linear operators, Part III: Spectral operators (1971) · Zbl 0243.47001 [8] Saphar, Bull. Soc. Math. France 92 pp 363– (1964) [9] DOI: 10.1007/BF02849344 · Zbl 0474.47008 · doi:10.1007/BF02849344 [10] DOI: 10.1007/BF02790238 · Zbl 0090.09003 · doi:10.1007/BF02790238 [11] Goldberg, Unbounded linear operators (1966) [12] DOI: 10.2307/2039803 · Zbl 0272.47020 · doi:10.2307/2039803 [13] Taylor, Introduction to functional analysis (1958) · Zbl 0081.10202
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