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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Similarity-invariant continuous functions on $ \mathcal{L}(\mathcal{H})$


Author: Domingo A. Herrero
Journal: Proc. Amer. Math. Soc. 97 (1986), 75-78
MSC: Primary 47A10
DOI: https://doi.org/10.1090/S0002-9939-1986-0831391-8
MathSciNet review: 831391
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Abstract: Let $ f:\mathcal{L}(\mathcal{H}) \to X$ be a continuous function from the algebra of all bounded linear operators acting on a complex infinite dimensional Hilbert space $ \mathcal{H}$ into a $ {T_1}$-topological space $ X$. If $ f(WA{W^{ - 1}}) = f(A)$ for all $ A$ in $ \mathcal{L}(\mathcal{H})$ and all invertible $ W$, then $ f$ is a constant function. The same result is true for a function $ f$ satisfying the above conditions defined on a connected open subset of $ \mathcal{L}{(\mathcal{H})_0} = \{ T \in \mathcal{L}(\mathcal{H}):T\,$has no normal eigenvalues.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0831391-8
Keywords: Similarity-invariant continuous function, closure of a similarity orbit, spectral functions, norm-topology, $ {T_1}$-topological space
Article copyright: © Copyright 1986 American Mathematical Society