Dunford–Schwartz theorem

In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L¹ converge in a suitable sense.[1]

Statement of the theorem

${\text{Let }}T{\text{ be a linear operator from }}L^{1}{\text{ to }}L^{1}{\text{ with }}\|T\|_{1}\leq 1{\text{ and }}\|T\|_{\infty }\leq 1{\text{. Then}}$

\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}T^{k}f

${\text{exists almost everywhere for all }}f\in L^{1}{\text{.}}$

The statement is no longer true when the boundedness condition is relaxed to even $\|T\|_{\infty }\leq 1+\varepsilon$ .[2]

Notes

Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", Journal of Rational Mechanics and Analysis, 5: 129–178, MR 0077090.
Friedman, N. (1966), "On the Dunford–Schwartz theorem", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 5 (3): 226–231, doi:10.1007/BF00533059, MR 0220900.

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[1] Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", Journal of Rational Mechanics and Analysis, 5: 129–178, MR 0077090.

[2] Friedman, N. (1966), "On the Dunford–Schwartz theorem", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 5 (3): 226–231, doi:10.1007/BF00533059, MR 0220900.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory