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Example of a mean ergodic L¹ operator with the linear rate of growth

100%

Kosek W.

Colloquium Mathematicum

2011

tom 124

nr 1

15-22

The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $lim_{n→ ∞}||Tⁿ||/n^{1-γ} = ∞$. In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $lim sup||Tⁿ||/n^{1-γ₀} = 0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible rate of growth, that is, $lim sup_{n → ∞}||Tⁿ||/n > 0$.

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

63%

Kornfeld I., Kosek W.

Colloquium Mathematicum

2003

tom 98

nr 1

63-77

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].

Ograniczanie wyników

2 Colloquium Mathematicum

2 Kosek W.

1 Kornfeld I.

1 2011

1 2003

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Example of a mean ergodic L¹ operator with the linear rate of growth

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille