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2003 | 98 | 1 | 63-77
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Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $||Tⁿ|| ∼ n^{1/4}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $lim_{n→ ∞} ||Tⁿ||/n^{1-γ} = ∞. In the class of positive operators these examples are the best possible in the sense that for every such operator T there exists a γ₀ > 0 such that $lim sup_{n→ ∞} ||Tⁿ||/n^{1-γ₀} = 0$.
A class of numerical sequences αₙ, intimately related to the problem of the growth of norms, is introduced, and it is shown that for every sequence αₙ in this class one can get ||Tⁿ|| ≥ αₙ (n = 1,2,...) for some T. Our examples can be realized in a class of positive L¹ operators associated with piecewise linear mappings of [0,1].
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  • Department of Mathematics, North Dakota State University, Fargo, ND 58105, U.S.A.
  • Department of Mathematics and Statistics, South Dakota State University, Brookings, SD 57007, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-1-5
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