algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
A bounded linear operator between Banach spaces is Fredholm if it has finite dimensional kernel and finite dimensional cokernel.
A parametrix of a bounded linear operator is a reverse operator which is an “inverse up to compact operators”, i.e. such that and are both compact operators.
(Atkinson 1951)
A bounded linear operator 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)
The original article:
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: -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]
Last revised on June 21, 2025 at 14:35:27. See the history of this page for a list of all contributions to it.