142 research outputs found

    Noncommutative Burkholder/Rosenthal inequalities II: applications

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    We show norm estimates for the sum of independent random variables in noncommutative LpL_p-spaces for 1<p<∞1<p<\infty following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the pp-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative LpL_p for 2<p<∞2<p<\infty.Comment: To appear in Isreal J; Mat

    Maximal theorems and square functions for analytic operators on Lp-spaces

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    Let T : Lp --> Lp be a contraction, with p strictly between 1 and infinity, and assume that T is analytic, that is, there exists a constant K such that n\norm{T^n-T^{n-1}} < K for any positive integer n. Under the assumption that T is positive (or contractively regular), we establish the boundedness of various Littlewood-Paley square functions associated with T. As a consequence we show maximal inequalities of the form \norm{\sup_{n\geq 0}\, (n+1)^m\bigl |T^n(T-I)^m(x) \bigr |}_p\,\lesssim\, \norm{x}_p, for any nonnegative integer m. We prove similar results in the context of noncommutative Lp-spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to R-boundedness properties

    Complex interpolation of weighted noncommutative LpL_p-spaces

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    Let M\mathcal{M} be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace τ\tau. Let dd be an injective positive measurable operator with respect to (M,τ)(\mathcal{M}, \tau) such that d−1d^{-1} is also measurable. Define Lp(d)=x∈L0(M):dx+xd∈Lp(M)and∥x∥Lp(d)=∥dx+xd∥p.L_p(d)={x\in L_0(\mathcal{M}) : dx+xd\in L_p(\mathcal{M})}\quad{and}\quad \|x\|_{L_p(d)}=\|dx+xd\|_p . We show that for 1\le p_0, 0<θ<10<\theta<1 and α0≥0,α1≥0\alpha_0\ge0, \alpha_1\ge0 the interpolation equality (Lp0(dα0),Lp1(dα1))θ=Lp(dα)(L_{p_0}(d^{\alpha_0}), L_{p_1}(d^{\alpha_1}))_\theta =L_{p}(d^{\alpha}) holds with equivalent norms, where 1p=1−θp0+θp1\frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1} and α=(1−θ)α0+θα1\alpha=(1-\theta)\alpha_0+\theta\alpha_1.Comment: To appear in Houston J. Mat

    A Helson-Szeg\"o theorem for subdiagonal subalgebras with applications to Toeplitz operators

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    We formulate and establish a noncommutative version of the well known Helson-Szego theorem about the angle between past and future for subdiagonal subalgebras. We then proceed to use this theorem to characterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated to subdiagonal subalgebras

    The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

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    We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let \M be a von Neumann algebra equipped with a normal faithful semifinite trace \t, and let EE be an r.i. space on (0, \8). Let E(\M) be the associated symmetric space of measurable operators. Then to any bounded linear map TT from E(\M) into a Hilbert space H\mathcal H corresponds a positive norm one functional f\in E_{(2)}(\M)^* such that \forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*), where E(2)E_{(2)} denotes the 2-concavification of EE and KK is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for E(\M) when EE is either 2-concave or 2-convex and qq-concave for some q<\8. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.Comment: 14 pages. To appear in J. Funct. Ana
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