Linear Maps Preserving the Set of Semi-Weyl Operators

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper, we characterized the linear maps ϕ:B(H)→B(H), which are surjective up to compact operators preserving the set of left semi-Weyl operators in both directions. As...

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Bibliographic Details
Published in:Mathematics (Basel) 2023-05, Vol.11 (9), p.2208
Main Authors: Yu, Wei-Yan, Cao, Xiao-Hong
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Calkin algebra
Food science
Hilbert space
left semi-Weyl operator
Linear operators
linear preservers
Operators (mathematics)
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Summary:Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper, we characterized the linear maps ϕ:B(H)→B(H), which are surjective up to compact operators preserving the set of left semi-Weyl operators in both directions. As an application, we proved that ϕ preserves the essential approximate point spectrum if and only if the ideal of all compact operators is invariant under ϕ and the induced map φ on the Calkin algebra is an automorphism. Moreover, we have ind(ϕ(T))=ind(T) if both ϕ(T) and T are Fredholm.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11092208