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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On unaveraged convergence of positive operators in Lebesgue space


Authors: H. Fong and L. Sucheston
Journal: Trans. Amer. Math. Soc. 179 (1973), 383-397
MSC: Primary 60F15; Secondary 28A65
MathSciNet review: 0329009
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Abstract: Let T be a power-bounded positive conservative operator on $ {L_1}$ of a $ \sigma $-finite measure space. Let e be a bounded positive function invariant under the operator adjoint to T.

Theorem. (1) $ \smallint \vert{T^n}f\vert \cdot \;e \to 0$ implies (2) $ \smallint \vert{T^n}f\vert \to 0$.

If T and all its powers are ergodic, and T satisfies an abstract Harris condition, then(l) holds by the Jamison-Orey theorem for all integrable f with $ \smallint f \cdot e = 0$, and hence also (2) holds for such f. A new proof of the Jamison-Orey theorem is given, via the 'filling scheme'. For discrete measure spaces this is due to Donald Ornstein, Proc. Amer. Math. Soc. 22 (1969), 549-551. If T is power-bounded, conservative and ergodic, and $ 0 < {f_0} = T{f_0}$, then $ {f_0}\; \cdot \;e \in {L_1}$ implies $ {f_0} \in {L_1}$, hence (2) implies that $ {T^n}f$ converges for each $ f \in {L_1}$.

Theorem. Let T be a positive conservative contraction on $ {L_1}$; then the class of functions $ \{ f - Tf,f \in L_1^ + \} $ is dense in the class of functions $ \{ f - Tf,f \in {L_1}\} $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0329009-1
PII: S 0002-9947(1973)0329009-1
Keywords: Positive operator, conservative and ergodic, invariant function, $ {L_1}$ convergence, transition measure, Harris condition
Article copyright: © Copyright 1973 American Mathematical Society