Norm and spectral radius for algebraic elements of a Banach algebra
1980 ◽
Vol 88
(1)
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pp. 129-133
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The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formulacontains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,(see Theorem 2), while for a general Banach algebra we have at least
1974 ◽
Vol 19
(1)
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pp. 59-69
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1979 ◽
Vol 22
(3)
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pp. 271-275
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Keyword(s):
1973 ◽
Vol 15
(2)
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pp. 129-137
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2019 ◽
Vol 11
(06)
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pp. 1950070
1968 ◽
Vol 8
(2)
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pp. 242-249
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Keyword(s):
1953 ◽
Vol 49
(1)
◽
pp. 59-62
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