nLab
Fredholm operator

Context

Functional analysis

Index theory

Contents

Definition

Definition

A continuous linear operator F:B 1B 2F \colon B_1\to B_2 between Banach spaces is Fredholm if it has finite dimensional kernel and finite dimensional cokernel.

Definition

The difference between the dimensions of the kernel and the cokernel of a Fredholm operator FF is called its index

indFdimkerFdimcokerF=dimkerFcodimimF. ind F \coloneqq dim ker F - dim coker F = dim ker F - codim im F \,.

A standard equivalent characterization of Fredholm operators is the following:

Definition

A parametrix of a bounded linear operator F: 1 2F \colon \mathcal{H}_1 \to \mathcal{H}_2 is a reverse operator P: 2 1P \colon \mathcal{H}_2 \to \mathcal{H}_1 which is an “inverse up to compact operators”, i.e. such that FPid 2F \circ P - id_{\mathcal{H}_2} and PFid 1P \circ F - id_{\mathcal{H}_1} are both compact operators.

Proposition

A bounded linear operator F:B 1B 2 F \colon B_1\to B_2 between Banach spaces is Fredholm, def. 1 precisely it is has a parametrix, def. 3.

Examples

Properties

Proposition

The image (range) of a Fredholm operator is closed.

Proposition

The subspace Fred(B 1,B 2)B(B 1,B 2)Fred(B_1,B_2)\subset B(B_1,B_2) of Fredholm operators in the space of bounded linear operators with the norm topology is open.

In other words, given a Fredholm operator FF, there exists ϵ>0\epsilon\gt 0 such that every bounded linear operator GG satisfying GF<ϵ\| G-F\|\lt \epsilon is Fredholm. Fredholm operators on a fixed separable Hilbert space H=B 1=B 2H = B_1 = B_2 form a semigroup with respect to the composition and the index is a morphism of semigroups: indFG=indF+indGind F G = ind F + ind G.

Proposition

The space FredFred of all Fredholm operators on an (infinite dimensional) separable Hilbert space is a model for the classifying space of degree-0 topological K-theory.

(…)

Generalizations

Fredholm operators generalize to Fredholm complexes. A finite chain complex

0C 0d 0C 1d 1C 2C n0 0 \to C_0 \stackrel{d_0}\to C_1\stackrel{d_1}\to C_2 \ldots C_n\to 0

of Banach spaces and bounded operators is said to be a Fredholm complex if the images d id_i are closed and the chain homology of the complex is finite dimensional. The Euler characteristic (the alternative sum of the dimensions of the homology groups) is then called the index of the Fredholm complex. Each Fredholm operator can be considered as a Fredholm complex concentrated at zero. Each Fredholm complex produces a Fredholm operator from the sum of the even- to the sum of the odd-numbered spaces in the complex.

One can consider Fredholm almost complexes, where d id i1d_i \circ d_{i-1} is not zero but the image of that operator is compact. Out of every Fredholm almost complex one can make a complex by correcting the differentials by compact perturbation terms. Elliptic complexes give examples of Fredholm complexes.

References

Related nnLab entries: Fredholm module, Fredholm determinant, Dirac operator, K-theory, KK-theory

  • wikipedia:Fredholm operator

  • A. S. Mishchenko, Векторные расслоения и их применения (Vector bundles and their applications), Nauka, Moscow, 1984. 208 pp. MR87f:55010

  • S. Rempel, B-W. Schulze, Index theory of elliptic boundary problems, Akademie–Verlag, Berlin, 1982.

  • Lars Hörmander, The analysis of linear partial differential operators. III. Pseudo-differential operators, 1994, reprinted 2007.

  • Pietro Aiena, Fredholm and local spectral theory, with applications to multipliers, book page

  • Otgonbayar Uuye, A simple proof of the Fredholm Alternative, arxiv/1011.2933

  • Alexander Grothendieck, La théorie de Fredholm, Bulletin de la Société Mathématique de France 84 (1956), p. 319-384, numdam

category: analysis

Revised on December 4, 2013 04:29:12 by Zoran Škoda (161.53.130.104)