Wyniki wyszukiwania

1

Pełczyński's Property (V) on spaces of vector-valued functions

100%

Randrianantoanina N.

Colloquium Mathematicum

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1996

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tom 71

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nr 1

63-78

2

Non-commutative martingale VMO-spaces

100%

Randrianantoanina N.

Studia Mathematica

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2009

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tom 191

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nr 1

39-55

EN

We study Banach space properties of non-commutative martingale VMO-spaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if ℳ is hyperfinite then the non-commutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of ℳ has property (u).

3

Factorization of operators on C*-algebras

100%

Randrianantoanina N.

Studia Mathematica

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1998

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tom 128

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nr 3

273-285

EN

Let A be a C*-algebra. We prove that every absolutely summing operator from A into $ℓ_2$ factors through a Hilbert space operator that belongs to the 4-Schatten-von Neumann class. We also provide finite-dimensional examples that show that one cannot replace the 4-Schatten-von Neumann class by the p-Schatten-von Neumann class for any p < 4. As an application, we show that there exists a modulus of capacity ε → N(ε) so that if A is a C*-algebra and $T ∈ Π_1(A,ℓ_2)$ with $π_1(T) ≤ 1$, then for every ε >0, the ε-capacity of the image of the unit ball of A under T does not exceed N(ε). This answers positively a question raised by Pełczyński.

4

Spectral subspaces and non-commutative Hilbert transforms

100%

Randrianantoanina N.

Colloquium Mathematicum

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2002

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tom 91

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nr 1

9-27

EN

Let G be a locally compact abelian group and ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. We study Hilbert transforms associated with G-flows on ℳ and closed semigroups Σ of Ĝ satisfying the condition Σ ∪ (-Σ) = Ĝ. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from L¹(ℳ,τ) into $L^{1,∞}(ℳ, τ)$. As an application, we obtain a Matsaev-type result for p = 1: if x is a quasi-nilpotent compact operator on a Hilbert space and Im(x) belongs to the trace class then the singular values ${μₙ(x)}_{n=1}^{∞}$ of x are O(1/n).

5

Noncommutative fractional integrals

64%

Randrianantoanina N., Wu L.

Studia Mathematica

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2015

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tom 229

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nr 2

113-139

EN

Let ℳ be a hyperfinite finite von Nemann algebra and $(ℳ_{k})_{k≥1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration $(ℳ_{k})_{k≥1}$. For a finite noncommutative martingale $x = (x_{k})_{1≤k≤ n} ⊆ L₁(ℳ)$ adapted to $(ℳ_{k})_{k≥1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{α}x = ∑_{k=1}^{n} ζ_{k}^{α} dx_{k}$ for an appropriate sequence $(ζ_{k})_{k≥1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁(𝓡) where 𝓡 is the type II₁ hyperfinite factor equipped with its natural increasing filtration, $ζ_{k} = 2^{-k}$ for k ≥ 1. We prove that $I^{α}$ is of weak type (1,1/(1-α)). More precisely, there is a constant c depending only on α such that if $x = (x_{k})_{k≥1}$ is a finite noncommutative martingale in L₁(ℳ) then $||I^{α}x||_{L_{1/(1-α),∞}(ℳ)} ≤ c||x||_{L₁(ℳ)}$. We also show that $I^{α}$ is bounded from $L_{p}(ℳ)$ into $L_{q}(ℳ)$ where 1 < p < q < ∞ and α = 1/p - 1/q, thus providing a noncommutative analogue of a classical result. Furthermore, we investigate the corresponding result for noncommutative martingale Hardy spaces. Namely, there is a constant ${c}$ depending only on α such that if $x = (x_{k})_{k≥1}$ is a finite noncommutative martingale in the martingale Hardy space 𝓗₁(ℳ) then $||I^{α}x||_{𝓗_{1/(1-α)}(ℳ)} ≤ c||x||_{𝓗₁(ℳ)}$.

Ograniczanie wyników

3 Studia Mathematica

2 Colloquium Mathematicum

5 Randrianantoanina N.

1 Wu L.

1 2015

1 2009

1 2002

1 1998

1 1996

Pełczyński's Property (V) on spaces of vector-valued functions

Non-commutative martingale VMO-spaces

Factorization of operators on C*-algebras

Spectral subspaces and non-commutative Hilbert transforms

Noncommutative fractional integrals