semi-Fredholm operator

A semi-Fredholm operator is a bounded operator between Banach spaces that has a finite dimensional kernel or cokernel, and closed range.There are two semigroups of semi-Fredholm operators:upper semi-Fredholm operators (those which have finite dimensional kernels),and lower semi-Fredholm operators (those which have finite dimensional cokernels).A semi-Fredholm operator that is both upper and lower is Fredholm.If an operator is homotopic through (a norm continuous path of) semi-Fredholm operators to a Fredholm operator, then it is Fredholm.