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

Mbekhta, Mostafa

Glasg. Math. J. 29, 159-175 (1987).

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

Keywords:

paranormal operator; semi-Fredholm operator; decomposition; Kato decomposition; spectral operator in the sense of Dunford

Citations:

Zbl 0090.090; Zbl 0474.47008

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