Abstract:

Let A be a bounded linear operator on a Hilbert space H. A∈B(H) is called a consistent in Fredholm (briefly CF operator) or A has property CF if σ_e(AB)=σ_e(BA) for every B∈B(H), where σ_e(·) denotes the essential spectrum. In this note, sufficient and necessary conditions for property CF are given, and the property CF for compact perturbation of A is considered. Finally, the properties CF of operator matrices are studied.

Key words: Fredholm operator; semi-Fredholm operator; spectrum