Fredholm Operators

One way of obtaining K-theory classes is to take a -term complex of vector bundles

we get get a K-theory class . In fact all K-theory classes occur like this if we allow to be infinite dimensional. This motivates the study of Fredholm operators.

Fix to be a separable Hilbert space over . [A vector space with an inner product that induces a complete metric on V].


Definition 13.2

Let be a bounded linear map (the image of the unit ball is bounded).

  1. We say that is Fredholm if both and are finite dimensional.
  2. The index of a Fredolm operator is defined .
  3. Fredholm operators .
 


Exercise 13.5

Consider the Hilbert space . The maps

have index respectively.

 
In fact  is a left inverse of . The operator  is the finite rank operator:

Constructing Index bundles[edit | edit source]

The theory of Fredholm operators is closely related to K-theory: Suppose that we had a smoothly varying family of Fredholm operators over a compact, connected space . Then such a family gives a K-theory class in , and all K-theory classes arise in this way.


Definition 13.3

Let be a family of Fredholm operators . Then we may associate a canonically defined K-theory class .

 
Proof

The map is continuous so is constant. In the simplified situation where and hence are also constant we can give an explicit description of the index bundles.

inherit the structure of vector bundles from the family. Define .

Now for the general situation consider the map (assuming for simplicity that ). Fredholm operators are invertible up to compact operators, so for each . By compactness of there is a global choice of that works for all . Now the family has Cokernel of constant dimension so apply the above construction. [HARD BIT: prove this is independent of ].

 


Exercise 13.6

Show that all K(X) classes arise this way by considering the sequence

[Here is the trivial vector bundle of sections of . The second map is ]

 


Exercise 13.7

Fredholm operators and winding numbers.

 

Let be the space of square integrable (complex) functions on . All such functions have a Fourier series

Define the Hardy space to be the subspace of functions with for all .

Let be a continuous function. Define an operator . Note that for . Define to be the projection of to by killing the negative Fourier coefficients.

is Fredholm . The nice fact about these Fredholm operators is that the index is the winding number of around .

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