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Invariance of spectrum for representations of |
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| Authors: | John Daughtry Alan Lambert Barnet Weinstock |
| Affiliation: | Department of Mathematics, East Carolina University, Greenville, North Carolina 27858 ; Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223 ; Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223 |
| Abstract: | Let  be a Banach space,  a unital  -algebra, and  an injective, unital homomorphism. Suppose that there exists a function  such that, for all  , and all  , (a) (b) (c) |
| Keywords: | |
| 点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息 | |
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, 
, 
. 
, the spectrum of 
in 
equals the spectrum of 
as a bounded linear operator on 
. If 
satisfies an additional requirement and 
is a 
-algebra, then the Taylor spectrum of a commuting 
-tuple 
of elements of 
equals the Taylor spectrum of the 
-tuple 
in the algebra of bounded operators on 
. Special cases of these results are (i) if 
is a closed subspace of a unital 
-algebra which contains 
as a unital 
-subalgebra such that 
, and 
only if 
, then for each 
, the spectrum of 
in 
is the same as the spectrum of left multiplication by 
on 
; (ii) if 
is a unital 
-algebra and 
is an essential closed left ideal in 
, then an element 
of 
is invertible if and only if left multiplication by 
on 
is bijective; and (iii) if 
is a 
-algebra, 
is a Hilbert 
-module, and 
is an adjointable module map on 
, then the spectrum of 
in the 
-algebra of adjointable operators on 
is the same as the spectrum of 
as a bounded operator on 
. If the algebra of adjointable operators on 
is a 
-algebra, then the Taylor spectrum of a commuting 
-tuple of adjointable operators on 
is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on 
.