On the Browder's theorem of an elementary operator

Abstract

Let H be an infinite complex Hilbert space and consider two bounded linear operators A, B ∈ L(H). Let Lᴀ ∈ L(L(H)) and Rʙ ∈ L(L(H)) be the left and the right multiplication operators, respectively, and denote by dᴀ;ʙ ∈ L(L(H)) either the elementary operator Δᴀ;ʙ (X) = (Lᴀ Rʙ ̶ I)(X) = A X B ̶̶ X or the generalized derivation δᴀ, ʙ (X) = (Lᴀ ̶ Rʙ)(X) = AX ̶ XB. This paper is concerned with the problem of the transference of Browder’s theorem from operators A and B to their elementary operator dᴀ,ʙ. We give necessary and sufficient conditions for dᴀ,ʙ to satisfy Browder’s theorem. Some applications for completely hereditarily normaloid operators are given.