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On an elementary operator with M ‐hyponormal operator entries
Author(s) -
Rashid M. H. M.
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300095
Subject(s) - mathematics , operator (biology) , hilbert space , weak operator topology , quasinormal operator , multiplication operator , pure mathematics , compact operator , shift operator , algebra over a field , finite rank operator , discrete mathematics , banach space , extension (predicate logic) , biochemistry , chemistry , repressor , computer science , transcription factor , gene , programming language
A Hilbert space operator T ∈ L ( H ) is M ‐hyponormal if there exists a positive real number M such that( T − μ )( T − μ ) * ≤ M 2( T − μ ) * ( T − μ )for all μ ∈ σ ( T ) . Let A , B * ∈ L ( H )be M ‐hyponormal and letd A B ∈ L ( L ( H ) )denote either the generalized derivationδ A B( X ) = A X − X B or the elementary operatorΔ A B = A X B − X . We prove that if A , B *are M ‐hyponormal, then f ( d A B ) satisfies the generalized Weyl's theorem and f ( d A B * ) satisfies the generalized a ‐Weyl's theorem for every f that is analytic on a neighborhood of σ ( d A B ) .