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

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

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

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

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