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

Let T bealinearoperatorfrom L ^{1} to L ^{1} with \|T\| _{1\leq} 1 and \|T\| _{infty\leq} 1 .Then \lim _{n → infty} (1)/(n) \sum ^{n -1} _{k =0} T ^{kf} existsalmosteverywhereforall f\in L ^{1.}

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

\|T\| _{infty\le} 1+ \varepsilon .^[2]

Notes and References

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