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

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

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

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

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