Linear Maps Preserving the Set of Semi-Weyl Operators

Wei-Yan Yu () and Xiao-Hong Cao
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Wei-Yan Yu: College of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
Xiao-Hong Cao: College of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062, China

Mathematics, 2023, vol. 11, issue 9, 1-7

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 ) , 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 i n d ( ϕ ( T ) ) = i n d ( T ) if both ϕ ( T ) and T are Fredholm.

Keywords: left semi-Weyl operator; Calkin algebra; linear preservers (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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