Left-Right Fredholm and Left-Right Browder Linear Relations

T. Álvarez, Fatma Fakhfakh, Maher Mnif

In this paper we introduce the notions of left (resp. right) Fredholm and left (resp. right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp. right) Browder linear relation $T$ in a Banach space as an operator-like sum $T=A+B$, where $A$ is an injective left (resp. a surjective right) Fredholm linear relation and $B$ is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp. right) Browder spectrum under finite rank operator.