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Linear Maps Preserving the Set of Semi-Weyl Operators

Author

Listed:
  • 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)

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.

Suggested Citation

  • Wei-Yan Yu & Xiao-Hong Cao, 2023. "Linear Maps Preserving the Set of Semi-Weyl Operators," Mathematics, MDPI, vol. 11(9), pages 1-7, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2208-:d:1141591
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