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 77090 .
↑ Friedman, N. (1966), "On the Dunford–Schwartz theorem", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 5 (3): 226–231, MR 220900, doi:10.1007/BF00533059 .

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