A GENERALISED COMMUTATIVITY THEOREM FOR P K-QUASIHYPONORMAL OPERATORS

For Hilbert space operators A and B, let δAB denote the generalised derivation δAB(X) = AX − XB and let △AB denote the elementary operator △AB(X) = AXB−X. If A is a pk-quasihyponormal operator, A ∈ pk − QH, and B ∗ is an either p-hyponormal or injective dominant or injective pk − QH operator (resp., B ∗ is an either p-hyponormal or dominant or pk − QH operator), then δAB(X) = 0 = ⇒ δA∗B∗(X) = 0 (resp., △AB(X) = 0 = ⇒ △A∗B∗(X) = 0)