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

Let $T$ be a linear operator from $L^1$ to $L^1$ with $\Vert T{\Vert }_1\le 1$ and $\Vert T{\Vert }_{\mathrm{\infty }}\le 1$. Then

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

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

The statement is no longer true when the boundedness condition is relaxed to even $\Vert T{\Vert }_{\mathrm{\infty }}\le 1+\epsilon$.

Notes

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