nLab Fredholm operator

Definition

Definition

A continuous linear operator $F \colon B_1\to B_2$ 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 $F$ is called its index (the Fredholm index)

$$ind F \coloneqq dim (ker F) - dim (coker F) = dim (ker F) - codim (im F) \,.$$

A standard equivalent characterization of Fredholm operators is the following:

Definition

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

Proposition

A bounded linear operator $F \colon B_1\to B_2$ between Banach spaces is Fredholm, def. precisely it is has a parametrix, def. .

Examples

• Elliptic operators on compact manifolds are naturally Fredholm, when understood between the appropriate Sobolev spaces.

• charged vacua of free Dirac field in Coulomb background are characterized by Fredholm operators

Properties

Proposition

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

Proposition

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

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

Proposition

The space $Fred$ 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

$$0 \to C_0 \stackrel{d_0}\to C_1\stackrel{d_1}\to C_2 \ldots C_n\to 0$$

of Banach spaces and bounded operators is said to be a Fredholm complex if the images $d_i$ 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 $d_i \circ d_{i-1}$ 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.

Related concepts

• Fredholm module,

• Fredholm determinant,

• Dirac operator,

• Toeplitz operator

• K-theory, KK-theory

• index theory

References

Textbook accounts:

• William Arveson, Section 3.3 of: A Short Course on Spectral Theory, Graduate Texts in Mathematics 209, Springer (2002) $[$doi:10.1007/b97227$]$

Discussion of the space of Fredholm operators as the classifying space for topological K-theory:

• Michael Atiyah, Appendix of: K-theory, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin 1967 (pdf, pdf)

See also:

• 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

• Marina Prokhorova, Spectral Sections, arXiv:2008.04672.

• Marina Prokhorova, Spaces of unbounded Fredholm operators. I. Homotopy equivalences, arXiv:2110.14359.

• Marina Prokhorova, The continuity properties of discrete-spectrum families of Fredholm operators, arXiv:2201.09869.

• Marina Prokhorova, From graph to Riesz continuity, arXiv:2202.03337.

For Fredholm complexes, see

• Graeme Segal, Fredholm complexes, Quarterly Journal of Mathematics 21:4 (1970), 385–402. doi.

category: analysis