142 research outputs found

### Noncommutative Burkholder/Rosenthal inequalities II: applications

We show norm estimates for the sum of independent random variables in
noncommutative $L_p$-spaces for $1<p<\infty$ following our previous work. These
estimates generalize the classical Rosenthal inequality in the commutative
case. Among applications, we derive an equivalence for the $p$-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 $L_p$ for $2<p<\infty$.Comment: To appear in Isreal J; Mat

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

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 $L_p$-spaces

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a
semifinite normal faithful trace $\tau$. Let $d$ be an injective positive
measurable operator with respect to $(\mathcal{M}, \tau)$ such that $d^{-1}$ is
also measurable. Define
$L_p(d)={x\in L_0(\mathcal{M}) : dx+xd\in
L_p(\mathcal{M})}\quad{and}\quad
\|x\|_{L_p(d)}=\|dx+xd\|_p .$ We show that for 1\le p_0,
$0<\theta<1$ and $\alpha_0\ge0, \alpha_1\ge0$ the interpolation equality
$(L_{p_0}(d^{\alpha_0}), L_{p_1}(d^{\alpha_1}))_\theta
=L_{p}(d^{\alpha})$ holds with equivalent norms, where
$\frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$ and
$\alpha=(1-\theta)\alpha_0+\theta\alpha_1$.Comment: To appear in Houston J. Mat

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

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

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 $E$ 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 $T$ from E(\M) into a Hilbert space $\mathcal H$
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
$E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant.
As a consequence we obtain the noncommutative Khintchine inequalities for
E(\M) when $E$ is either 2-concave or 2-convex and $q$-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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