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On the axiomatic theory of spectrum II

100%
Mbekhta M., Müller V.
Studia Mathematica
|
1996
|
tom 119
|
nr 2
129-147
EN
We give a survey of results concerning various classes of bounded linear operators in a Banach space defined by means of kernels and ranges. We show that many of these classes define a spectrum that satisfies the spectral mapping property.
2
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Closed range multipliers and generalized inverses

100%
B. Laursen K., Mbekhta M.
Studia Mathematica
|
1993
|
tom 107
|
nr 2
127-135
EN
Conditions involving closed range of multipliers on general Banach algebras are studied. Numerous conditions equivalent to a splitting A = TA ⊕ kerT are listed, for a multiplier T defined on the Banach algebra A. For instance, it is shown that TA ⊕ kerT = A if and only if there is a commuting operator S for which T = TST and S = STS, that this is the case if and only if such S may be taken to be a multiplier, and that these conditions are also equivalent to the existence of a factorization T = PB, where P is an idempotent and B an invertible multiplier. The latter condition establishes a connection to a famous problem of harmonic analysis.
3
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Corrigendum and addendum: "On the axiomatic theory of spectrum II"

81%
Koliha J. J., Mbekhta M., Müller V., Poon P. W.
Studia Mathematica
|
1998
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tom 130
|
nr 2
193-198
EN
The main purpose of this paper is to correct the proof of Theorem 15 of [4], concerned with the stability of the class of quasi-Fredholm operators under finite rank perturbations, and to answer some open questions raised there.
4
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Bounded point evaluations for multicyclic operators

81%
EL Guendafi M., Mbekhta M., Zerouali E.
Banach Center Publications
|
2005
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tom 67
|
nr 1
199-217
EN
Let T be a multicyclic operator defined on some Banach space. Bounded point evaluations and analytic bounded point evaluations for T are defined to generalize the cyclic case. We extend some known results on cyclic operators to the more general setting of multicyclic operators on Banach spaces. In particular we show that if T satisfies Bishop's property (β), then $ℬ_a = ℬ ∖ σ_{ap}(T)$. We introduce the concept of analytic structures and we link it to different spectral quantities. We apply this concept to retrieve in an easy way a theorem of D. Herrero and L. Rodman: the set of cyclic n-tuples for a multicyclic operator T is dense if and only if $ℬ_a = ∅$.
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