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Proceedings of the American Mathematical Society

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On Akcoglu and Sucheston's operator convergence theorem in Lebesgue space


Author: Ryōtarō Satō
Journal: Proc. Amer. Math. Soc. 40 (1973), 513-516
MSC: Primary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1973-0341138-0
MathSciNet review: 0341138
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Abstract: Let $ T$ be a bounded linear operator on an $ {L_1}$-space and $ \tau $ its linear modulus. It is proved that if the adjoint of $ \tau $ has a strictly positive subinvariant function then the following two conditions are equivalent: (i) $ {T^n}$ converges weakly; (ii) $ (1/n)\Sigma _{i = 1}^n{T^{{k_i}}}$ converges strongly for any strictly increasing sequence $ {k_1},{k_2}, \cdots $ of nonnegative integers.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0341138-0
Keywords: Bounded linear operator, linear modulus of a bounded linear operator, weak and strong convergence
Article copyright: © Copyright 1973 American Mathematical Society