“…Moreover, for the results of our work, there is no loss of generality in assuming that U' 0 = /, the identity operator, where U' 0 belongs to the restriction of ^to # [2,3]. Hence one obtains the conclusions of Theorem 3.5 and Theorem 3.6 on fé 7 . It is obvious that in general there is no convergence on Sd.…”
Section: F ^ F -\-Ii F + Ijmentioning
confidence: 69%
“…If {~F( M ,v))( w ,v)ec is ^-superadditive, then {^( MiV) } (w , v)ec is called ^-subadditive; and' if both {-i 7 (M , v) } (M , v)ec and {^( M , v) } (M , v)GC are ^-superadditive, then {L V )L v)e( ; is called ^-additive [2,3].…”
Section: ~Kmentioning
confidence: 99%
“…Any strongly ^-superadditive process F = {^W V )L v)É(: which satisfies (1)(2)(3)(4)(5)(6)(7)(8) *io,v> = F («,0) = °> «>0, v>0…”
The ergodic theorem for multidimensional strongly subadditive processes relative to a semigroup induced by a measure preserving point transformation on X was proved by R. T. Smythe [18]. His results have been generalized by M. A. Akçoğlu and U. Krengel [4] to the continuous parameter case. The definition of superadditivity they used is stronger than Smythe's but weaker than strong superadditivity. R. Emilion and B. Hachem [10] extended this result to strongly superadditive processes relative to a semigroup generated by a pair of commuting Markovian operators which are also L∞-contractions. The basic tool in the proof is a technique which may be referred to as “reduction of dimension“ and they used a version of it due to A. Brunei [6].The purpose of this paper is to show that if F = {F(uv)}u>0 is a bounded strongly superadditive process with respect to a two-dimensional strongly continuous Markovian semigroup of operators on L1, then u-2F(uu) converges a.e. as u → ∞.
“…Moreover, for the results of our work, there is no loss of generality in assuming that U' 0 = /, the identity operator, where U' 0 belongs to the restriction of ^to # [2,3]. Hence one obtains the conclusions of Theorem 3.5 and Theorem 3.6 on fé 7 . It is obvious that in general there is no convergence on Sd.…”
Section: F ^ F -\-Ii F + Ijmentioning
confidence: 69%
“…If {~F( M ,v))( w ,v)ec is ^-superadditive, then {^( MiV) } (w , v)ec is called ^-subadditive; and' if both {-i 7 (M , v) } (M , v)ec and {^( M , v) } (M , v)GC are ^-superadditive, then {L V )L v)e( ; is called ^-additive [2,3].…”
Section: ~Kmentioning
confidence: 99%
“…Any strongly ^-superadditive process F = {^W V )L v)É(: which satisfies (1)(2)(3)(4)(5)(6)(7)(8) *io,v> = F («,0) = °> «>0, v>0…”
The ergodic theorem for multidimensional strongly subadditive processes relative to a semigroup induced by a measure preserving point transformation on X was proved by R. T. Smythe [18]. His results have been generalized by M. A. Akçoğlu and U. Krengel [4] to the continuous parameter case. The definition of superadditivity they used is stronger than Smythe's but weaker than strong superadditivity. R. Emilion and B. Hachem [10] extended this result to strongly superadditive processes relative to a semigroup generated by a pair of commuting Markovian operators which are also L∞-contractions. The basic tool in the proof is a technique which may be referred to as “reduction of dimension“ and they used a version of it due to A. Brunei [6].The purpose of this paper is to show that if F = {F(uv)}u>0 is a bounded strongly superadditive process with respect to a two-dimensional strongly continuous Markovian semigroup of operators on L1, then u-2F(uu) converges a.e. as u → ∞.
“…The converse is not so straightforward and needs the full extent of the hypothesis, that is, that (II) holds for any f . Chacon described in [1] an example of a positive contraction U on some L 1 (X, µ) and a positive bounded and integrable function f such that…”
Given a space X, a σ-algebra B on X and a measurable map T : X → X, we say that a measure µ is half-invariant if, for any B ∈ B, we have µ(T −1 (B) ≤ µ(B).In this note we present a generalization of Birkhoff's Ergodic theorem to σ-finite half-invariant measures.
“…Example 1. In [2] or [4, Example 6.8] a general method is given for constructing ergodic transformations that do not accept a finite invariant measure and in [ll] or [4, Example 6.8] a conservative and ergodic measure preserving transformation on a probability space is constructed having no eigenvalues except 1. It is possible to combine these two techniques and obtain an ergodic transformation that does not accept a finite invariant measure and also that has no eigenvalues except 1.…”
ABSTRACT, Aj is an ergodic conservative transformation on a 0"-finite measure space and A2 is an ergodic measure preserving transformation on a finite measure space. We study the point spectrum properties of Aj X ¿2' 1°p articular we show Aj x A2 is ergodic if and only if Aj x A2 have no eigenvalues in common other than the eigenvalue 1. The conditions on Aj, h2 stated above are in a sense the most general for the validity of this result.
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