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

Satō, Ryōtarō

doi:10.1090/S0002-9939-1973-0341138-0

On Akcoglu and Sucheston’s operator convergence theorem in Lebesgue space
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by Ryōtarō Satō

Proc. Amer. Math. Soc. 40 (1973), 513-516

DOI: https://doi.org/10.1090/S0002-9939-1973-0341138-0

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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

M. Akcoglu and L. Sucheston, On operator convergence in Hilbert space and in Lebesgue space, Period. Math. Hungar. 2 (1972), 235–244. MR 326433, DOI 10.1007/BF02018664
R. V. Chacon and U. Krengel, Linear modulus of linear operator, Proc. Amer. Math. Soc. 15 (1964), 553–559. MR 164244, DOI 10.1090/S0002-9939-1964-0164244-7
Ry\B{o}tar\B{o} Sat\B{o}, Ergodic properties of bounded $L_{1}$-operators, Proc. Amer. Math. Soc. 39 (1973), 540–546. MR 414828, DOI 10.1090/S0002-9939-1973-0414828-9