Fredholm operator - Knowlpedia

Let $H_0,H_1$ be Hilbert spaces. A bounded operator $T:H_0\to H_1$ is Fredholm if its range is closed and both $\ker T$ and the cokernel $H_1/\operatorname{ran}T$ are finite-dimensional. Equivalently, $\ker T$ and $\ker T^*$ are finite-dimensional and $\operatorname{ran}T$ is closed. Its Fredholm index is

\operatorname{ind}(T)=\dim\ker T-\dim\ker T^*.

The closed-range hypothesis is essential: finite-dimensional kernel and dense, nonclosed range do not make an operator Fredholm.

Atkinson characterization

Atkinson's theorem says that $T$ is Fredholm exactly when it is invertible modulo compact operators. Concretely, this means there is a bounded $S:H_1\to H_0$ such that

ST-I_{H_0} \quad\text{and}\quad TS-I_{H_1}

are compact. Such an $S$ is called a parametrix modulo compact operators. When $H_0=H_1=H$ , the image of $T$ is therefore invertible in the Calkin algebra $B(H)/K(H)$ .

Stability and index

Fredholmness is open in the operator norm, and the index is locally constant on the set of Fredholm operators. If $K:H_0\to H_1$ is compact, then

T+K\ \text{is Fredholm} \qquad\text{and}\qquad \operatorname{ind}(T+K)=\operatorname{ind}(T).

If $T:H_0\to H_1$ and $S:H_1\to H_2$ are Fredholm, then

\operatorname{ind}(ST)=\operatorname{ind}(S)+\operatorname{ind}(T).

These facts make the index insensitive to many analytic perturbations while retaining global information.

Example

The unilateral shift $U:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ ,

U(x_0,x_1,\ldots)=(0,x_0,x_1,\ldots),

has zero-dimensional kernel and a one-dimensional cokernel. Hence $U$ is Fredholm and $\operatorname{ind}(U)=-1$ .

Unbounded convention

A closed densely defined operator is also sometimes called Fredholm when it has closed range and finite-dimensional kernel and cokernel. That is a distinct unbounded notion: boundedness, domains, and adjoints must be handled explicitly. Under suitable regularity, an unbounded Fredholm operator can be represented by a bounded transform, but the two definitions should not be silently identified.

References