Fredholm Operators

Fredholm Operators

Fredholm Operator

Let be a Hilbert space. A bounded linear operator is said to be Fredholm if:

1. is a closed subspace of .
2. is finite-dimensional.
3. is finite-dimensional.

Index of a Fredholm Operator

The index of a Fredholm operator is defined as the integer: Since is closed, we have and . Thus, , and the index can be written as:

e.g.

• Let be the unilateral shift operator on . Then is Fredholm and .
• If an operator is invertible, then it is Fredholm and .
• If is a compact operator, then is a Fredholm operator.

Atkinson's Lemma

An operator is Fredholm if and only if it is invertible modulo the compact operators. That is, there exist operators such that: for some compact operators .

Proof () Suppose such and exist.

1. Range: . Since is Fredholm, its range is closed and has finite codimension. Therefore, must also be closed and have finite codimension.
2. Kernel: . Since is compact, , which implies is finite-dimensional.

Thus, is Fredholm. () If is Fredholm, its restriction is a continuous bijection , and thus has a continuous inverse . Extend to all of by defining it to be on . One can then verify that and . Since and , the projection operators and are finite-rank and therefore compact.

The Calkin Algebra

Calkin Algebra

The set of all compact operators on a Hilbert space forms a closed, two-sided -ideal in the C*-algebra of bounded operators . The quotient algebra is called the Calkin algebra. We denote the canonical quotient map by .

Corollary

An operator is Fredholm if and only if its image is invertible in the Calkin algebra .

Properties of the Index

Index of a Product

If are Fredholm operators, then the product is also Fredholm, and its index is the sum of the indices:

Topological Properties

The set of Fredholm operators is an open subset of . The index map is continuous.

Proof The Calkin algebra is a Banach algebra, so its group of invertible elements is open. Since and the quotient map is continuous, is the continuous preimage of an open set and is therefore open. To show continuity of the index, we can show it is locally constant. For any , there exists a neighborhood where the index is constant. If is an inverse of modulo compacts, then for any with , the operator is invertible. Since invertible operators have index 0, we get , which implies .

Invariance under Compact Perturbations

If is a Fredholm operator and is a compact operator, then is also Fredholm and:

Proof Consider the path defined by . The map is continuous, and the index map is continuous, so their composition is continuous. Since is connected, its image must be a connected subset of . The only connected subsets of are single points, so must be a constant function. Therefore, .

Relationship with Toeplitz Operators

Proposition

A Toeplitz operator with continuous symbol is Fredholm if and only if its symbol is invertible (i.e., for all ).

Proof is Fredholm iff is invertible in the Calkin algebra. There exists a -isomorphism from onto a subalgebra of the Calkin algebra that maps . By spectral permanence, is invertible in the Calkin algebra iff it is invertible in this subalgebra , which is true iff is invertible in .

Lemma

If is an invertible function, then there exists a unique integer such that can be written in the form: where and . Moreover, this is the winding number of .

Index of a Toeplitz Operator

If is invertible, then the Toeplitz operator is Fredholm and its index is given by the negative of the winding number of its symbol:

Proof First, one shows that if and , then is invertible, which implies . Now, let be any invertible symbol. By the lemma, we can write , where . The operator can be shown to be compact. By the invariance of index under compact perturbations and the index of a product property, we have: Since is invertible, its index is 0. A direct calculation shows . Therefore,

e.g. Suppose . Note that has winding number and has winding number (because it has a pole of order one at , and a zero of order one at , so the argument principle applies). Thus, , and the index of the Toeplitz operator is .