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.

Statement of the theorem

$\text{Let }T \text{ be a linear operator from }L^1 \text{ to }L^1 \text{ with }\|T\|_1\leq 1\text{ and }\|T\|_\infty\leq 1 \text{. Then}$

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}T^kf$

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

The statement is no longer true when the boundedness condition is relaxed to even $\|T\|_\infty\le 1+\varepsilon$.