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A counterexample of hermitian liftings

Published online by Cambridge University Press:  20 January 2009

Pei-Kee Lin
Pei-Kee Lin
Affiliation:
Department of Mathematical Sciences, Memphis State University, Memphis, TN 38152, U.S.A.
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Let X be a complex Banach space, and let and denote respectively the algebras of bounded and compact operators on X. The quotient algebra is called the Calkin algebra associated with X. It is known that both and are complex Banach algebras with unit e. For such unital Banach algebras B, set

and define the numerical range of x ∈ B as

x is said to be hermitian if W(x)⊆R. It is known that

Fact 1. ([4 vol. I, p. 46]) x is hermitian if and only if ‖eiαx‖ = (or ≦)1 for all α ∈ R, where ex is defined by

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 2 , June 1989 , pp. 255 - 259
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

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