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Keywords = Calkin algebra

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Article
Linear Maps Preserving the Set of Semi-Weyl Operators
by Wei-Yan Yu and Xiao-Hong Cao
Mathematics 2023, 11(9), 2208; https://doi.org/10.3390/math11092208 - 8 May 2023
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Abstract
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) [...] Read more.
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. Full article
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