ON AN ELEMENTARY OPERATOR WITH p-w-HYPONORMAL OPERATOR ENTRIES

A Hilbert space operatorA∈L(H) isp-w-hyponormal,A∈pw−H, if|eT|p≥ |T|p≥ |eT∗|pfor 0< p≤1:LetdAB=ABor ∆AB, whereAB∈L(L(H)) is the generalized derivationAB(X) =AX−XBand ∆AB∈L(L(H)) is the elementary operator ∆AB(X) =AXB−X:We prove that ifA; B∗∈pw−H, thendABhas the single-valued extension property, and thequasinilpotent partH0(dAB−) ofdABat∈iso(dAB) equals (dAB−)−1(0):Also, we provethat ifA; B∗∈pw−H, thenf(dAB) satisfies Weyl’s theorem for each functionfthat is analyticon a neighborhood of(dAB):