Left-Right Fredholm and Left-Right Browder Linear Relations

Abstract

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.