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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
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.

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

  • [1] M. Akcoglu and L. Sucheston, On operator convergence in Hilbert space and in Lebesgue space, Periodica Math. Hungarica 2 (1972), 235-244. MR 0326433 (48:4777)
  • [2] R. V. Chacon and U. Krengel, Linear modulus of a linear operator, Proc. Amer. Math. Soc. 15 (1964), 553-559. MR 29 #1543. MR 0164244 (29:1543)
  • [3] R. Sato, Ergodic properties of bounded $ {L_1}$-operators, Proc. Amer. Math. Soc. 39 (1973), 540-546. MR 0414828 (54:2920)

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Keywords: Bounded linear operator, linear modulus of a bounded linear operator, weak and strong convergence
Article copyright: © Copyright 1973 American Mathematical Society

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