nLab Atkinson's theorem

Preliminaries

Definition

A bounded 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

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.

Statement

Theorem

(Atkinson 1951)
A bounded linear operator $F \colon B_1\to B_2$ between Banach spaces is Fredholm, def. , precisely if it admits a parametrix, def. .

(cf. Pedersen 1989 Prop. 3.3.11, Murphy 1990 Thm. 1.4.16)

References

The original article:

• F. V. Atkinson: The normal solubility of linear equations in normed spaces, Sb. Math. 70 (1951) [mathnet:sm5589]

Review:

• Martin Schechter: Basic theory of Fredholm operators, Annali della Scuola Normale Superiore di Pisa - Scienze Fisiche e Matematiche, Serie 3, 21 2 (1967) 261-280 [numdam:ASNSP_1967_3_21_2_261_0]

• Gert K. Pedersen: Analysis Now, Graduate Texts in Mathematics 118, Springer (1989) [doi;10.1007/978-1-4612-1007-8]

• Gerard Murphy: §1.4 in: $C^\ast$-algebras and Operator Theory, Academic Press (1990) [doi:10.1016/C2009-0-22289-6]

See also:

• Wikipedia: Atkinson’s theorem

• Ethan Y. Jaffe: Atkinson’s Theorem [pdf]