An alternating procedure for operators on $L_ p$ spaces

Abstract:

Let ${L_p}$ be the usual Banach spaces over a $\sigma$-finite measure space. If $1{\text { < }}p{\text { < }}\infty$ and $q = p{(p - 1)^{ - 1}}$, then ${\psi _p}:{L_p} \to {L_q}$ denotes the duality mapping defined by the requirements that $(f,{\psi _p}f) = \left \| f \right \|_p^p = {\left \| f \right \|_p}\left \| {{\psi _p}f} \right \|q,f \in {L_p}$. If $T:{L_p} \to {L_p}$ is a bounded linear operator, then $M(T):{L_p} \to {L_p}$ is the mapping defined by $M(T) = {\psi _q}{T^ * }{\psi _p}T$, where ${T^ * }:{L_q} \to {L_q}$ is the adjoint of $T$. It is proved that if ${T_n}$ is a sequence of operators on ${L_p}$ such that $\left \| {{T_n}} \right \| \leq 1$ for all $n$, then $M({T_n} \cdots {T_2}{T_1})f$ converges in ${L_p}$ for all $f \in {L_p}$.