On an elementary operator with w-hyponormal operator entries

Let A,B∗∈B(H) be w-hyponormal operators, and let dAB∈B(B(H)) denote either the generalized derivation δAB(X)=AX-XB or the length two elementary operator ▵AB(X)=AXB-X. We prove that dAB has the single–valued extension property, and the quasinilpotent part H0(dAB-λ) of dAB at equals (dAB-λ)-1(0). Let H(σ(dAB)) denote the space of functions which are analytic on σ(dAB), and let Hc(σ(dAB)) denote the space of f∈H(σ(dAB)) which are non-constant on every connected component of σ(dAB). It is proved that, for every h∈H(σ(dAB)) and f,g∈Hc(σ(dAB)), the complement of the Weyl spectrum σw(h(df(A)g(B))) of h(df(A)g(B)) in σ(h(df(A)g(B))) consists of isolated points in σ(h(df(A)g(B))) which are eigenvalues of finite multiplicity.