Wyniki wyszukiwania

1

Vector-valued ergodic theorems for multiparameter Additive processes II

100%

Sato R.

Colloquium Mathematicum

|

2003

|

tom 97

|

nr 1

117-129

EN

Previously we obtained stochastic and pointwise ergodic theorems for a continuous d-parameter additive process F in L₁((Ω,Σ,μ);X), where X is a reflexive Banach space, under the condition that F is bounded. In this paper we improve the previous results by considering the weaker condition that the function $W(·) = ess sup{||F(I)(·)||: I ⊂ [0, 1)^{d}}$ is integrable on Ω.

2

Pointwise ergodic theorems for functions in Lorentz spaces $L_{pq}$ with p ≠ ∞

100%

Sato R.

Studia Mathematica

|

1994

|

tom 109

|

nr 2

209-216

EN

Let τ be a null preserving point transformation on a finite measure space. Assuming τ is invertible, P. Ortega Salvador has recently obtained sufficient conditions for the almost everywhere convergence of the ergodic averages in $L_{pq}$ with 1 < p < ∞, 1 < q < ∞. In this paper we obtain necessary and sufficient conditions for the almost everywhere convergence, without assuming that τ is invertible and only assuming that p ≠ ∞.

3

On a vector-valued local ergodic theorem in $L_∞$

100%

Sato R.

Studia Mathematica

|

1999

|

tom 132

|

nr 3

285-298

EN

Let $T = {T(u): u ∈ ℝ_d^{+}}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_1((Ω,Σ,μ);X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_1((Ω,Σ,μ);X)* = L_∞((Ω,Σ,μ);X*)$, the adjoint semigroup $T* = {T*(u): u ∈ ℝ_d^{+}}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_∞((Ω,Σ,μ);X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u ∈ ℝ_d^{+}$, has a contraction majorant P(u) defined on $L_1((Ω,Σ,μ);ℝ)$, that is, P(u) is a positive linear contraction on $L_1((Ω,Σ,μ);ℝ)$ such that $‖T(u)f(ω)‖ ≤ P(u)‖f(·)‖(ω)$ almost everywhere on Ω for every $⨍ ∈ L_1((Ω,Σ,μ);X)$, we prove that the local ergodic theorem holds for T*.

4

On invariant measures for power bounded positive operators

100%

Sato R.

Studia Mathematica

|

1996

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tom 120

|

nr 2

183-189

EN

We give a counterexample showing that $\overline{(I-T*)L_{∞}} ∩ L^{+}_{∞} = {0}$ does not imply the existence of a strictly positive function u in $L_1$ with Tu = u, where T is a power bounded positive linear operator on $L_1$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.

5

Differentiation of Banach-space-valued additive processes

100%

Sato R.

Studia Mathematica

|

2001

|

tom 147

|

nr 2

131-153

EN

Let X be a Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let L be a Banach space of X-valued strongly measurable functions on (Ω,Σ,μ). We consider a strongly continuous d-dimensional semigroup $T = {T(u): u = (u₁,...,u_{d})$, $u_{i} > 0$, 1 ≤ i ≤ d} of linear contractions on L. We assume that each T(u) has, in a sense, a contraction majorant and that the strong limit $T(0) = strong-lim_{u→0} T(u)$ exists. Then we prove, under some suitable norm conditions on the Banach space L, that a differentiation theorem holds for d-dimensional bounded processes in L which are additive with respect to the semigroup T. This generalizes a differentiation theorem obtained previously by the author under the assumption that L is an X-valued $L_{p}$-space, with 1 ≤ p < ∞.

6

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

100%

Sato R.

Studia Mathematica

|

1995

|

tom 114

|

nr 3

227-236

EN

Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{-1} ∑^{n-1}_{i=0} f∘τ^{i}(x)$ converges almost everywhere to a function f* in $L(p_1,q_1]$, where (pq) and $(p_1,q_1]$ are assumed to be in the set ${(r,s) : r=s=1, or 1 < r < ∞ and 1 ≤ s ≤ ∞, or r = s = ∞}$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified

7

On solvability of the cohomology equation in function spaces

100%

Sato R.

Studia Mathematica

|

2003

|

tom 156

|

nr 3

277-293

EN

Let T be an endomorphism of a probability measure space (Ω,𝓐,μ), and f be a real-valued measurable function on Ω. We consider the cohomology equation f = h ∘ T - h. Conditions for the existence of real-valued measurable solutions h in some function spaces are deduced. The results obtained generalize and improve a recent result of Alonso, Hong and Obaya.

8

Solvability of the functional equation f = (T-I)h for vector-valued functions

100%

Sato R.

Colloquium Mathematicum

|

2004

|

tom 99

|

nr 2

253-265

EN

Let X be a reflexive Banach space and (Ω,𝓐,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,𝓐,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $lim_{n→∞} fₙ = f$ in measure for some f ∈ M(μ;X), then also $lim_{n→∞} Tfₙ = Tf$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying f = (T-I)h.

9

Vector-valued ergodic theorems for multiparameter additive processes

100%

Sato R.

Colloquium Mathematicum

|

1999

|

tom 79

|

nr 2

193-202

EN

Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T={T(u):u=($u_{1}$, ... ,$u_{d})$, $u_{i}$ ≥ 0, 1 ≤ i ≤ d } be a strongly measurable d-parameter semigroup of linear contractions on $L_{1}$((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_{1}$((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_{1}$((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems for a d-parameter bounded additive process F in $L_{1}$((Ω,Σ,μ);X) with respect to the semigroup T.

10

A general differentiation theorem for multiparameter additive processes

100%

Sato R.

Colloquium Mathematicum

|

2002

|

tom 91

|

nr 1

143-155

EN

Let $(L,||·||_{L})$ be a Banach lattice of equivalence classes of real-valued measurable functions on a σ-finite measure space and $T = {T(u): u = (u₁,...,u_{d}), $u_{i} > 0$, 1 ≤ i ≤ d}$ be a strongly continuous locally bounded d-dimensional semigroup of positive linear operators on L. Under suitable conditions on the Banach lattice L we prove a general differentiation theorem for locally bounded d-dimensional processes in L which are additive with respect to the semigroup T.

11

A general differentiation theorem for superadditive processes

100%

Sato R.

Colloquium Mathematicum

|

2000

|

tom 83

|

nr 1

125-136

EN

Let L be a Banach lattice of real-valued measurable functions on a σ-finite measure space and T={$T_t$: t < 0} be a strongly continuous semigroup of positive linear operators on the Banach lattice L. Under some suitable norm conditions on L we prove a general differentiation theorem for superadditive processes in L with respect to the semigroup T.

12

A note on the almost everywhere convergence of alternating sequences with Dunford-Schwartz operators

100%

Sato R.

Colloquium Mathematicum

|

1991

|

tom 62

|

nr 1

97-101

13

A weighted ergodic maximal equality for nonsingular semiflows

64%

Ephremidze L., Sato R.

Colloquium Mathematicum

|

2005

|

tom 103

|

nr 2

207-213

EN

A weighted ergodic maximal equality is proved for a conservative and ergodic semiflow of nonsingular automorphisms.

14

Boundedness and growth orders of means of discrete and continuous semigroups of operators

52%

Li Y.-C., Sato R., Shaw S.-Y.

Studia Mathematica

|

2008

|

tom 187

|

nr 1

1-35

EN

We discuss implication relations for boundedness and growth orders of Cesàro means and Abel means of discrete semigroups and continuous semigroups of linear operators. Counterexamples are constructed to show that implication relations between two Cesàro means of different orders or between Cesàro means and Abel means are in general strict, except when the space has dimension one or two.

15

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Operators on continuous function spaces and L1-spaces

44%

Sato R.

Colloquium Mathematicum

|

1975-1976

|

tom 34

|

nr 2

257-269

16

A simple proof of the Chui-Smith theorem on Landau's problem

38%

Sato M., Sato R.

Colloquium Mathematicum

|

1982

|

tom 47

|

nr 1

119-122

17

On the ergodic power function for invertible positive operators

38%

Sato R.

Studia Mathematica

|

1988

|

tom 90

|

nr 2

129-134

18

A mean ergodic theorem for a contraction semigroup in Lebesgue space

38%

Sato R.

Studia Mathematica

|

1975-1976

|

tom 54

|

nr 3

213-219

19

Operators on some function spaces

32%

Sato R.

Colloquium Mathematicum

|

1976

|

tom 36

|

nr 1

117-120

20

On the individual ergodic theorem for subsequences

32%

Sato R.

Studia Mathematica

|

1973

|

tom 45

|

nr 1

31-35

Ograniczanie wyników

Vector-valued ergodic theorems for multiparameter Additive processes II

Pointwise ergodic theorems for functions in Lorentz spaces $L_{pq}$ with p ≠ ∞

On a vector-valued local ergodic theorem in $L_∞$

On invariant measures for power bounded positive operators

Differentiation of Banach-space-valued additive processes

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

On solvability of the cohomology equation in function spaces

Solvability of the functional equation f = (T-I)h for vector-valued functions

Vector-valued ergodic theorems for multiparameter additive processes

A general differentiation theorem for multiparameter additive processes

A general differentiation theorem for superadditive processes

A note on the almost everywhere convergence of alternating sequences with Dunford-Schwartz operators

A weighted ergodic maximal equality for nonsingular semiflows

Boundedness and growth orders of means of discrete and continuous semigroups of operators

Operators on continuous function spaces and L1-spaces

A simple proof of the Chui-Smith theorem on Landau's problem

On the ergodic power function for invertible positive operators

A mean ergodic theorem for a contraction semigroup in Lebesgue space

Operators on some function spaces

On the individual ergodic theorem for subsequences