A bound for the Hilbert–Schmidt norm of generalized commutators of nonself–adjoint operators

Let A, Ã and B be bounded linear operators in a Hilbert space, and f (z) be a function regular on the convex hull of the union of the spectra of A and Ã. Let SN₂ be the ideal of Hilbert-Schmidt operators. In the paper, a sharp estimate for the Hilbert-Schmidt norm of the commutator f(A)B−B f(Ã) is established, provided AB−BÃ ∈ SN₂ , A−A∗ ∈ SN₂ and Ã−Ã ∗ ∈ SN₂. Here the star means the adjointness. Our results are new even in the finite dimensional case.