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.^[1]

Theorem. Let T be a linear operator from L₁ to L₁ with $\|T\|_1\leq 1$ and $\|T\|_\infty\leq 1$. Then

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}T^kf$

exists almost everywhere for all $f\in L_1$.

The statement is no longer true when the boundedness condition is relaxed to even $\|T\|_\infty\le 1+\varepsilon$.^[2]

See also

• Bartle–Dunford–Schwartz theorem

Notes

1. ^ Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", J. Rational Mech. Anal. 5: 129–178, MR77090 .
2. ^ Friedman, N. (1966), "On the Dunford–Schwartz theorem", Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (3): 226–231, doi:10.1007/BF00533059, MR220900 .

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