Estimates of positive linear operators on $L^ p$

Abstract:

Let $0 < \alpha < 1$ and $T$ a positive linear operator on ${L^p},1 < p < + \infty$, such that ${|| {(1 - \alpha )I + \alpha T} ||_p} \leq 1$ ($I$ = identity). For such operators, (which do not necessarily satisfy (i) ${|| T ||_p} \leq 1$ (contraction), (ii) ${\sup _{n \geq 1}}{\left \| {(I + T + \cdots + {T^{n - 1}})/n} \right \|_p} \leq 1$ (Cesàro mean bounded by one)) [1] we show, using M. A. Akcoglu’s estimate, that \[ {\left \| {\sup \limits _{\begin {array}{*{20}{c}} {n \geq 1} \\ {n \in N} \\ \end {array} } \frac {{f + Tf + \cdots + {T^{n - 1}}f}}{n}} \right \|_p} \leq \gamma (\alpha )||f|{|_p}\;{\text {for any }}f \in {L^p}.\] We also obtain the pointwise ergodic theorem in ${L^p}$.