[Paper Review] Random Series of Trace Class Operators

Gilles Pisier|arXiv (Cornell University)|Mar 10, 2011

Advanced Banach Space Theory16 references3 citations

TL;DR

This paper investigates the almost sure convergence of random series of trace class operators using Gaussian and Rademacher sequences, establishing a probabilistic framework that links convergence in $L_p$-norms to almost sure convergence. The key contribution is a characterization of the spaces $G(B)$ and $R(B)$—spaces of almost surely convergent series—via tensor product decompositions and the analytic UMD property, particularly showing $R(E\widehat{\otimes}F) \subset R(E)\widehat{\otimes}F + E\widehat{\otimes}R(F)$ under suitable conditions.

ABSTRACT

In this lecture, we present some results on Gaussian (or Rademacher) random series of trace class operators, mainly due jointly with F. Lust-Piquard. We will emphasize the probabilistic reformulation of these results, as well as the open problems suggested by them. We start by a brief survey of what is known about the problem of characterizing a.s. convergent (Gaussian or Rademacher) series of random vectors in a Banach space. The main result presented here is that for certain pairs of Banach spaces $E,F$ that include Hilbert spaces (and type 2 spaces with the analytic UMD property), we have $$ R(E\hat\otimes F) =R(E)\hat\otimes F + E\hat\otimes R(F) $$ where $R(E)$ denotes the space of convergent Rademacher series with coefficients in $E$ and $E\hat\otimes F$ denotes the projective tensor product.

Motivation & Objective

• To characterize the almost sure convergence of random series $\sum g_n x_n$ and $\sum \varepsilon_n x_n$ in Banach spaces $B$ using probabilistic and functional analytic tools.
• To identify the Banach spaces $G(B)$ and $R(B)$—spaces of almost surely convergent series—as isomorphic to tensor products $L_p(\mu; \ell_2)$ or $H^1$-based spaces under specific conditions.
• To establish a derivation-like formula for the Rademacher series space $R$ on operator spaces, showing $R(E\widehat{\otimes}F) \subset R(E)\widehat{\otimes}F + E\widehat{\otimes}R(F)$.
• To explore open problems regarding higher-order tensor products and second derivatives of $R$-spaces, particularly in the case $E=F=G=\ell_2$.

Proposed method

• Uses the equivalence of almost sure convergence and $L_p$-convergence for $0 \leq p < \infty$, reducing the problem to $L_2$-norm estimates.
• Applies Talagrand’s majorizing measure condition as a general solution to Problem 1, but focuses on concrete Banach spaces with additional structure (e.g., $L_p$, Hilbert, operator spaces).
• Employs the analytic UMD property and Haar-type decompositions to represent functions in $H^1(\Delta, B)$ via $g_m, h_m$ and extract series representations.
• Introduces a complex Steinhaus series $f(t_0,t_1,\ldots) = \sum e^{it_n} x_n$ and uses integration over the infinite torus $\Delta$ to relate $L_1$-norms to Rademacher series norms.
• Derives norm inequalities via $L_2$-estimates on $H^2(\Delta, E)$ and $H^2(\Delta, F)$, leading to bounds on $\|\sum \varepsilon_n x_n\|_{R(E\widehat{\otimes}F)}$.
• Uses the structure of the infinite-dimensional torus and conditional expectations $E_n$ to decompose $x_n$ into components involving $d_n g_m$ and $d_n h_m$, enabling tensor decomposition.

Experimental results

Research questions

• RQ1 Under what conditions on a sequence $(x_n)$ in a Banach space $B$ does the random series $\sum g_n x_n$ converge almost surely in norm?
• RQ2 Can the space $R(E\widehat{\otimes}F)$ of almost surely convergent Rademacher series on a tensor product be characterized as a sum of $R(E)\widehat{\otimes}F$ and $E\widehat{\otimes}R(F)$?
• RQ3 Is the space $L_p\widehat{\otimes}L_q$ of finite cotype for $1 < p, q < 2$, and what is its cotype when $2 \leq p, q < \infty$?
• RQ4 Does a derivation-like formula extend to higher-order tensor products, such as $R(E\widehat{\otimes}F\widehat{\otimes}G)$, yielding a sum of $R$-terms on each factor?
• RQ5 Can a second derivative formula be established for $R_1(R_2(E\widehat{\otimes}F))$, analogous to the product rule in calculus?

Key findings

• The spaces $G(B)$ and $R(B)$ of almost surely convergent Gaussian and Rademacher series are isomorphic to $L_p(\mu; \ell_2)$ when $B = L_p(\mu)$, with equivalent norms depending only on $p$.
• For $B = L_p(\mu)$ with $1 \leq p < \infty$, the series $\sum \varepsilon_n x_n$ converges a.s. if and only if $\int \left( \sum |x_n(s)|^2 \right)^{p/2} d\mu(s) < \infty$.
• The space $R(E\widehat{\otimes}F)$ is contained in $R(E)\widehat{\otimes}F + E\widehat{\otimes}R(F)$, with the norm of the inclusion bounded by a constant depending on the UMD constants $K(E)$ and $K(F)$.
• The norm of $\sum \varepsilon_n x_n$ in $R(E\widehat{\otimes}F)$ is bounded by $4(CK(E) + CK(F)) \|S\|_{R(E\widehat{\otimes}F)}$, proving the inclusion with controlled constants.
• For $2 \leq p, q < \infty$, the space $L_p\widehat{\otimes}L_q$ is of cotype $\max(p,q)$, extending known results for $p=q=2$.
• The open problem remains whether $L_p\widehat{\otimes}L_q$ has finite cotype (e.g., cotype 2) for $1 < p < 2$, $1 < q \leq 2$, and whether $\ell_2\widehat{\otimes}\ell_2\widehat{\otimes}\ell_2$ has finite cotype.