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 of the theorem

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

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

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

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