Dunford–Schwartz theorem

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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 L_1 to L_1 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]

Notes[edit]

  1. ^ Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", J. Rational Mech. Anal. 5: 129–178, MR 77090 .
  2. ^ Friedman, N. (1966), "On the Dunford–Schwartz theorem", Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (3): 226–231, doi:10.1007/BF00533059, MR 220900 .