Dunford–Schwartz theorem explained

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

LetTbealinearoperatorfromL1toL1with\|T\|1\leq1and\|T\|infty\leq1.Then

\limn → infty

1
n
n-1
\sum
k=0

Tkf

existsalmosteverywhereforallf\inL1.

The statement is no longer true when the boundedness condition is relaxed to even

\|T\|infty\le1+\varepsilon

.[2]

Notes and References

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