Fredholm Operator - Properties

Properties

The set of Fredholm operators from X to Y is open in the Banach space L(X, Y) of bounded linear operators, equipped with the operator norm. More precisely, when T₀ is Fredholm from X to Y, there exists ε > 0 such that every T in L(X, Y) with ||T − T₀|| < ε is Fredholm, with the same index as that of T₀.

When T is Fredholm from X to Y and U Fredholm from Y to Z, then the composition is Fredholm from X to Z and

When T is Fredholm, the transpose (or adjoint) operator T ′ is Fredholm from Y ′ to X ′, and ind(T ′) = −ind(T). When X and Y are Hilbert spaces, the same conclusion holds for the Hermitian adjoint T∗.

When T is Fredholm and K a compact operator, then T + K is Fredholm. The index of T remains constant under compact perturbations of T. This follows from the fact that the index i(s) of T + s K is an integer defined for every s in, and i(s) is locally constant, hence i(1) = i(0).

Invariance by perturbation is true for larger classes than the class of compact operators. For example, when T is Fredholm and S a strictly singular operator, then T + S is Fredholm with the same index. A bounded linear operator S from X to Y is strictly singular when its restriction to any infinite dimensional subspace X₀ of X fails to be an into isomorphism, that is: