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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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