On an elementary operator with M ‐hyponormal operator entries

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 ) .