Dunford–Schwartz theorem - Alchetron, the free social encyclopedia

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

Let T be a linear operator from L 1 to L 1 with ∥ T ∥ 1 ≤ 1 and ∥ T ∥ ∞ ≤ 1 . Then

lim n → ∞ 1 n ∑ k = 0 n − 1 T k f

exists almost everywhere for all f ∈ L 1 .

The statement is no longer true when the boundedness condition is relaxed to even ∥ T ∥ ∞ ≤ 1 + ε .

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