Paranormal Operators and Some Operator Equations

Abstract

Let a pair (A, B) of bounded linear operators acting on a Hilbert space be a solution of the operator equations ABA = A(2) and BAB = B-2. When A is a paranormal operator, we explore some behaviors of the operators AB, BA, and B. In particular, if A or A* is a polynomial root of paranormal operators, we show that Weyl type theorems are satisfied for the operators AB, BA, and B.