Dunford–Schwartz theorem
From HandWiki
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 of the theorem
[math]\displaystyle{ \text{Let }T \text{ be a linear operator from }L^1 \text{ to } L^1 \text{ with } \|T\|_1\leq 1\text{ and }\|T\|_\infty\leq 1 \text{. Then} }[/math]
- [math]\displaystyle{ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}T^kf }[/math]
[math]\displaystyle{ \text{exists almost everywhere for all }f\in L^1\text{.} }[/math]
The statement is no longer true when the boundedness condition is relaxed to even [math]\displaystyle{ \|T\|_\infty\le 1+\varepsilon }[/math].[2]
Notes
- ↑ Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", Journal of Rational Mechanics and Analysis 5: 129–178.
- ↑ Friedman, N. (1966), "On the Dunford–Schwartz theorem", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 5 (3): 226–231, doi:10.1007/BF00533059.
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