Lacunary matrices

任意非交换$L_1$空间的2-共缺子序列原理

The $2$-co-lacunary subsequence principle for arbitrary noncommutative $L_1$-spaces

该研究针对任意非交换$L_1$空间，证明其Haagerup空间中有界序列必含范数收敛或2-共缺子序列，解决了公开问题并放宽了条件。

2-Co-lacunary sequences in noncommutative symmetric Banach spaces

We characterize noncommutative symmetric Banach spaces for which every bounded sequence admits either a convergent subsequence, or a $2$-co-lacunary subsequence. This extends the classical characterization, due to Räbiger.

Lacunary series and stable distributions

By well known results of probability theory, any sequence of random variables with bounded second moments has a subsequence satisfying the central limit theorem and the law of the iterated logarithm in a randomized form. In this paper we give criteria for a sequence $(X_n)$ of random variables to have a subsequence $(X_{n_k})$ whose weighted partial sums, suitably normalized, converge weakly to a stable distribution with parameter $0<α<2$.

Paley's theorem for Hankel matrices via the Schur test

Paley's theorem about lacunary coefficients of functions in the classical space $H^1$ on the unit circle is equivalent to the statement that certain Hankel matrices define bounded operators on $\ell^2$ of the nonnegative integers. Since that statement reduces easily to the case where the entries in the matrix are all nonnegative, it must be provable by the Schur test. We give such proofs with interesting patterns in the vectors used in the test, and we recover the best constant in the main case. We use related ideas to reprove the characterization of Paley multipliers from $H^1$ to $H^2$.

Cycles and 1-unconditional matrices

We characterize the 1-unconditional subsequences of the canonical basis (e_rc) of elementary matrices in the Schatten-von-Neumann class S^p . The set I of couples (r,c) must be the set of edges of a bipartite graph without cycles of even length 4<=l<=p if p is an even integer, and without cycles at all if p is a positive real number that is not an even integer. In the latter case, I is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space S^p_I spanned by (e_rc)_{(r,c)\in I} in S^p .