THE FUGLEDE-PUTNAM THEOREM FOR (p,k)-QUASIHYPONORMAL OPERATORS

We show that if T ∈() is a (p,k)-quasihyponormal operator and S ∗ ∈() is a p-hyponormal operator, and if TX = XS, where X: → is a quasiaffinity (i.e., a one-one map having dense range), then T is a normal and moreover T is unitarily equivalent to S. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. Let be a separable complex Hilbert space with inner product 〈·,· 〉 and let () denote the C∗-algebra of all bounded linear operators on . The spectrum of an operator T, denoted by σ(T), is the set of all complex numbers λ for which T − λI is not invertible. The numerical range of an operator T, denoted by W(T), is the set defined by W(T) = {〈Tx,x 〉 : ‖x ‖ = 1}. (1) The norm closure of a subspace of is denoted by . We denote the kernel and the range of an operator T by ker(T) and ran(T), respectively. For p such as 0 < p ≤ 1 and positive integer k, an operator T ∈() is called (p,k)-quasihyponormal if T∗k(|T|2p−|T∗|2p)Tk ≥ 0. A (p,k)-quasihyponormal operator is an extension of p-hyponormal operator (i.e., (T∗T)p−(TT∗)p ≥ 0), k-quasihyponormal operator (i.e., T∗k(|T|2−|T∗|2)Tk ≥ 0) and p-quasihyponormal operator (i.e., T∗(|T|2