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 Let T {\textstyle T} be a linear operator from L 1 {\textstyle L^{1}} to L 1 {\textstyle L^{1}} with ‖ T ‖ 1 ≤ 1 {\textstyle \|T\|_{1}\leq 1} and ‖ T ‖ ∞ ≤ 1 {\textstyle \|T\|_{\infty }\leq 1} . Then
lim n → ∞ 1 n ∑ k = 0 n − 1 T k f {\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}T^{k}f}
exists almost everywhere for all f ∈ L 1 {\textstyle f\in L^{1}} . The statement is no longer true when the boundedness condition is relaxed to even ‖ T ‖ ∞ ≤ 1 + ε {\displaystyle \|T\|_{\infty }\leq 1+\varepsilon } .
Notes