On a theorem of Gelfand and its local generalizations

Driss Drissi

Studia Mathematica (1997)

  • Volume: 123, Issue: 2, page 185-194
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Drissi, Driss. "On a theorem of Gelfand and its local generalizations." Studia Mathematica 123.2 (1997): 185-194. <http://eudml.org/doc/216387>.

@article{Drissi1997,
abstract = {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.},
author = {Drissi, Driss},
journal = {Studia Mathematica},
keywords = {locally power-bounded operator; local spectrum; local spectral radius; doubly power-bounded element of a Banach algebra; Bernstein inequality for multivariable functions; Gelfand’s theorem for commuting bounded operators},
language = {eng},
number = {2},
pages = {185-194},
title = {On a theorem of Gelfand and its local generalizations},
url = {http://eudml.org/doc/216387},
volume = {123},
year = {1997},
}

TY - JOUR
AU - Drissi, Driss
TI - On a theorem of Gelfand and its local generalizations
JO - Studia Mathematica
PY - 1997
VL - 123
IS - 2
SP - 185
EP - 194
AB - 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.
LA - eng
KW - locally power-bounded operator; local spectrum; local spectral radius; doubly power-bounded element of a Banach algebra; Bernstein inequality for multivariable functions; Gelfand’s theorem for commuting bounded operators
UR - http://eudml.org/doc/216387
ER -

References

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