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On the axiomatic theory of spectrum

100%
Kordula V., Müller V.
Studia Mathematica
|
1996
|
tom 119
|
nr 2
109-128
EN
There are a number of spectra studied in the literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra, which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semiregular or essentially semiregular).
2
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On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space

100%
Drissi D.
Studia Mathematica
|
1998
|
tom 130
|
nr 1
1-7
EN
Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.
3
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Local spectrum and Kaplansky's theorem on algebraic operators

100%
Drissi D.
Colloquium Mathematicum
|
1998
|
tom 75
|
nr 2
159-165
EN
Using elementary arguments we improve former results of P. Vrbová concerning local spectrum. As a consequence, we obtain a new proof of Kaplansky’s theorem on algebraic operators on a Banach space.
4
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On a theorem of Gelfand and its local generalizations

100%
Drissi D.
Studia Mathematica
|
1997
|
tom 123
|
nr 2
185-194
EN
In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = {1}, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille's results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand's theorem for m commuting bounded operators, $T_1,..., T_m$, on a Banach space X.
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