Abstract
The present paper is concerned with the study of a new class of linear operators on a Hilbert space: the class of quasi-Fredholm operators, which contains many operators already studied in the litterature (in particular semi-Fredholm operators). An operatorA is said to be quasi-Fredholm of degreed, if the following conditions are satisfied:
-
a)
For alln greater thand, R(A n )∩N(A)=R(A d )∩N(A);
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b)
N(A)∩R(A d) is closed inH;
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c)
R(A)+N(A d) is closed inH.
Two characterisations of quasi-Fredholm operators are given:
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1)
A is quasi-Fredholm iff there exists a direct decomposition ofH into the sum of two subspacesH 1 andH 2 which are invariant underA and such that the restriction ofA toH 1 is quasi-Fredholm of degree 0 and the restriction ofA toH 2 is nilpotent (Kato decomposition).
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2)
A is quasi-Fredholm iff there exists a neighborhoodD of 0 in C such that for all λ≠0 in that neighborhoodA−λI has a generalized inverse which is meromorphic inD−{0} (The generalized inverse is holomorphic inD iffA is of degree 0).
The bulk of the paper is devoted to the proofs of these characterizations and of related results, making use of the theory of operators ranges and of generalized inverses. Most of the results extend easily to the Banach case.
The rest of the paper deals with the class of quasi-normal operators, which is closely related to the class of spectral operators. Some applications of the first part of the paper are given in this context.
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References
Banach S.,Théorie des opérations linéaires, Mon. Mat., Varsovie 1932.
Bart H.,Holomorphic relative inverses of operator valued functions, Math. Annalen,208 (1974), 179–194.
Bart H.,Poles of the resolvent of an operator function, Proc. Roy. Ir. Acad.,74A (1974), 169–184.
Bart H.-Kaballo W.,Local invertibility of meromorphic operator functions (à paraître dans Proc. Roy. Ir. Acad.).
Bart H.-Kaashoek M., A.-Lay D. C.,Relative inverses of meromorphic operator functions, Univ. of Maryland Techn. Report TR 74-71 (1974).
Bart H.-Kaashoek M. A.-Lay D. C.,Relative inverses of meromorphic operator functions and associated holomorphic projection functions, Math. Annalen,218 (1975), 199–210.
Bart H.-Lay D. C.,Poles of a generalized resolvent operator, Proc. Roy. Irish Acad.,74A (1974), 147–168.
Caradus S. R.,Operators of Riesz type, Pac. Jour. Math.,18(1) (1966), 61–71.
Caradus S. R.,On meromorphic operators (I) et (II), Can. Jour. Math.,19 (1967).
Caradus S. R.,Operator theory of the pseudo-inverse, Queen's papers in pure and applied mathematics n. 38. Queen university, Kingston, Ontario 1974.
Caradus S. R.,Generalized inverses and operator theory, Technical report (à paraitre).
Caradus S. R.-Pfaffenberger W. E.-Yood B.,Calkin algebras and algebra of operators on Banach spaces, Lecture notes in pure and applied mathematics, Marcel Dekker Inc New York 1974.
Colojoara I.-Foias C.,Theory of generalized spectral operators, Gordon and Breach, New York 1968.
Cordes H. O.-Labrousse J. P.,The invariance of the Index in the Metric space of Closed Operators, Jour. of Math. and Mech.,12 (1963), 693–720.
Dixmier J.,Etude sur les variétés et les opérateurs de Julia, Bull. Soc. Math. France,77 (1949), 11–101.
Dunford N.-Schwartz J.,Linear Operators Part III, Wiley, Interscience, New York 1971.
Fillmore P.-Williams J.,On operators ranges, Advance in Math.,7 (1971), 254–282.
Förster K. H.-Kaashoek M. A.,The asymptotic behaviour or the reduced minimum modulus of a Fredholm operator, Proc. A.M.S.,49 (1975), 123–131.
Gokhberg I. C.,Quelques propriétés d'opérateurs normalement solubles (en Russe), Dok. Akad. Nauk SSSR (N. S.),104 (1955), 9–11.
Gokhberg I. C.-Markus A. S.,Propriétés caractéristiques de certains points du spectre d'opérateurs linéaires bornés (en Russe), Izv. Utch. Zaved. Matematika,2 (1960), 74–78.
Goldberg S.,Unbounded Linear Operators, New York, Mc Graw Hill 1966.
Goldman M. A.-Kratchowsky S. N.,Sur la stabilité de quelques propriétés d'un opérateur linéaire fermé, (en Russe), Dok. Akad. Nauk SSSR,209 (1973) (traduction anglaise: Sov. Math. Dokl.,14 (1973) n. 2).
Grabiner S.,Operators with eventual uniform ascent and descent, Technical report Pomona college, Claremont, California.
Grabiner S.,Operators with almost uniform ascent and descent, Technical report Pomona college, Claremont, California.
Kaashoek M. A.,Stability theorems for closed linear operators, Proc. Acad. Sci. Amsterdam A,68 (1965), 452–466.
Kaashoek M. A.,Ascent, descent, nullity and defect, Math. Annalen,172 (1967), 105–115.
Kaashoek M. A.,On the Riesz set of a linear operator, Nederl. Akad. Wetensh. Proc. Ser. A,71—Indag. Math,30 (1968), 46–53.
Kato T.,Perturbation theory for linear operators, Springer Verlag, Berlin 1966.
Kato T.,Perturbation theory for nullity, deficiency, and other quantities of linear operators, Jour. Anal. Math.,6 (1958), 261–322.
Labrousse J. Ph.,Une caractérisation topologique des générateurs infinitésimaux de semi-groupes analytiques et de contraction sur un espace de Hilbert, Acad. Naz. dei Lincei, (8)52.
Labrousse J. Ph.,On a metric space of closed operators on a Hilbert space, Rev. Mat. y Fis. T. Ser. A Univ. Nat. de Tucumán (Argentine),16 (1966), 45–77.
Labrousse J. Ph.,Conditions nécessaires et suffisantes pour qu'un opérateur soit décomposable au sens de Kato, C. R. Acad. Sci. Paris,284 (1977), 295–298.
Labrousse J. Ph.,Opérateurs spectraux et opérateurs quasi-normaux, C. R. Acad. Sci. Paris,286 (1978), 1107–1108.
Lay D. C.,Spectral analysis using ascent, descent, nullity and defect, Math., Annalen,184 (1970), 197–214.
Neubauer G.,Espaces Paracomplets, Conférence à Nice, juin 1974.
Saphar P.,Contribution à l'étude des applications linéaires dans un espace de Banach, Bull. Soc. Math. France,92 (1964), 363–384.
Shubin M. A.,Familles holomorphes de sous-espaces d'un espace de Banach (en Russe), Math. Issled.,5 (1970), 153–165.
Taylor A. E.,Theorems on ascent, descent, nullity and defect of linear operators, Math. Annalen,163 (1966), 18–49.
West T. T.,Riesz operators in Banach spaces, Proc. Lond. Math. Soc., (3)16 (1966), 131–140.
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Labrousse, JP. Les operateurs quasi Fredholm: Une generalisation des operateurs semi Fredholm. Rend. Circ. Mat. Palermo 29, 161–258 (1980). https://doi.org/10.1007/BF02849344
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DOI: https://doi.org/10.1007/BF02849344