Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
MathSciNet review: 958065
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

  • [1] M. A. Akcoglu, A pointwise ergodic theorem in 𝐿_{𝑝}-spaces, Canad. J. Math. 27 (1975), no. 5, 1075–1082. MR 0396901 (53 #761)
  • [2] I. Assani, Sur la théorie ergodique des opérateurs dans les espaces $ {L^p}$, Séminaire d'Initiation à l'Analyse (G. Choquet, J. St. Raymond, M. Rogalski) Communication 1, 1984-1985.
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • [4] R. Émilion, On the pointwise ergodic theorems in 𝐿_{𝑝}\(1<𝑝<∞), Studia Math. 81 (1985), no. 2, 171–178. MR 818179 (87c:47009)
  • [5] Ryōtarō Satō, Ergodic theorems for semi-groups in 𝐿_{𝑝}, 1<𝑝<∞, Tôhoku Math. J. (2) 26 (1974), 73–76. MR 0343064 (49 #7808)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 46E30, 47A35

Retrieve articles in all journals with MSC: 47B38, 46E30, 47A35


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958065-9
PII: S 0002-9939(1988)0958065-9
Article copyright: © Copyright 1988 American Mathematical Society