The Fredholm spectrum of the sum and product of two operators

Abstract:

Let $C(X)$ denote the set of closed operators with dense domain on a Banach space X, and $L(X)$ the set of all bounded linear operators on X. Let ${\mathbf {\Phi }}(X)$ denote the set of all Fredholm operators on X, and ${\sigma _{\mathbf {\Phi }}}(A)$ the set of all complex numbers ${\mathbf {\lambda }}$ such that $({\mathbf {\lambda }} - A) \notin {\mathbf {\Phi }}(X)$. In this paper we establish conditions under which ${\sigma _{\mathbf {\Phi }}}(A + B) \subseteq {\sigma _{\mathbf {\Phi }}}(A) + {\sigma _{\mathbf {\Phi }}}(B),{\sigma _{\mathbf {\Phi }}}(\overline {BA} ) \subseteq {\sigma _{\mathbf {\Phi }}}(A) \cdot {\sigma _{\mathbf {\Phi }}}(B)$, and ${\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)$.