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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An alternating procedure for operators on $L_ p$ spaces
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by M. A. Akcoglu and L. Sucheston
Proc. Amer. Math. Soc. 99 (1987), 555-558
DOI: https://doi.org/10.1090/S0002-9939-1987-0875396-0

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 $f \in {L_p}$.
References
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Bibliographic Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 99 (1987), 555-558
  • MSC: Primary 47A35; Secondary 28D99, 47B38
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0875396-0
  • MathSciNet review: 875396