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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Estimates of positive linear operators on $ L\sp p$


Author: I. Assani
Journal: Proc. Amer. Math. Soc. 104 (1988), 193-196
MSC: Primary 47B38; Secondary 46E30, 47A35
DOI: https://doi.org/10.1090/S0002-9939-1988-0958065-9
MathSciNet review: 958065
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Abstract: Let $ 0 < \alpha < 1$ and $ T$ a positive linear operator on $ {L^p},1 < p < + \infty $, such that $ {\vert\vert {(1 - \alpha )I + \alpha T} \vert\vert _p} \leq 1$ ($ I$ = identity). For such operators, (which do not necessarily satisfy

(i) $ {\vert\vert T \vert\vert _p} \leq 1$ (contraction),

(ii) $ {\sup _{n \geq 1}}{\left\Vert {(I + T + \cdots + {T^{n - 1}})/n} \right\Vert _p} \leq 1$ (Cesàro mean bounded by one)) [1] we show, using M. A. Akcoglu's estimate, that

$\displaystyle {\left\Vert {\mathop {\sup }\limits_{\begin{array}{*{20}{c}} {n \... ...\leq \gamma (\alpha )\vert\vert f\vert{\vert _p}\;{\text{for any }}f \in {L^p}.$

We also obtain the pointwise ergodic theorem in $ {L^p}$.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0958065-9
Article copyright: © Copyright 1988 American Mathematical Society