On invariant measures for power bounded positive operators

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Title

Authors ¹

Department of Mathematics, Faculty of Science, Okayama University, Okayama, 700 Japan

We give a counterexample showing that

\bar{(I - T \cdot) L_{\infty}} \cap L^{+}_{\infty} = {0}

does not imply the existence of a strictly positive function u in

L_{1}

with Tu = u, where T is a power bounded positive linear operator on

L_{1}

of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.

power bounded and Cesàro bounded positive operators, invariant measures,

L_{1}

spaces

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