On the operator equations ABA = A2 and BAB = B2

Abstract

Abstract. We generalize a result of I. Vidav concerning the operator equations ABA = A 2 and BAB = B 2 .

On the Dunford Property (C) for Bounded Linear Operators RS and SR

“…In this case R, S, SR and RS share many spectral properties [9,10], and local spectral properties as decomposability, property (β) and SVEP [6]. In this section we consider the permanence of property (C) in this context.…”

A note on the common spectral properties for bounded linear operators

“…In the following two examples, the common spectral properties for AC and BA can only followed directly from the above results, but not from the corresponding ones in [7,9,15,16,19].…”

Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

“…An element a ∈ R is said to be Vidav [10] proved that A and B are self-adjoint operators satisfying operator equations ABA = A 2 and BAB = B 2 if and only if A = F F * and B = F * F for some idempotent operator F . Schmoeger [8,9] and Duggal [7] also obtained the common spectral properties of bounded linear operators A and B satisfying operator equations ABA = A 2 and BAB = B 2 . Zeng and Zhong [12] proved that AC is Drazin invertible (polaroid) if and only if BA is Drazin invertible (polaroid) for operator equation ABA = ACA.…”