On cotype and summing properties in Banach spaces.

zbMATH Open1071.47023arXivmath/9312206MaRDI QIDQ1850246

Publication date: 1 January 2003

Published in: Illinois Journal of Mathematics (Search for Journal in Brave)

Abstract: It is well known in Banach space theory that for a finite dimensional space

E

there exists a constant

c_{E}

, such that for all sequences

(x_{k})_{k} s u b s e t E

one has [ summ_k

oo x_k rm kl c_E pl sup_{eps_k pm 1} oo summ_k eps_k x_k rm pl .] Moreover, if $E$ is of dimension $n$ the constant $c_{E}$ ranges between $s q r t n$ and $n$ . This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces $X$ , where the constant $c_{E} s i m s q r t n$ for all finite dimensional subspaces. More generally, we prove that an estimate $c_{E} k l l c n^{1 - f r a c 1 q}$ holds for all $n i n z$ and all $n$ -dimensional subspaces $E$ of $X$ if and only if the eigenvalues of every operator factoring through $e l l_{i n f t y}$ decrease of order $k^{- f r a c 1 q}$ if and only if $X$ is of weak cotype $q$ , introduced by Pisier and Mascioni. We emphasize that in contrast to Talagrand's equivalence theorem on cotype $q$ and absolutely $(q, 1)$ -summing spaces this extendsto the case $q = 2$ . If $q > 2$ and one of the conditions above is satisfied one has [ kla summ_k oo x_k rm^q mer^{frac{1}{q}} kl C^{1+l}pl (1+{ m log}_2)^{(l)}((1 +{ m log}_2 n)^{frac{1}{q}}) pl ez

oo summ_k eps_k x_k rm ] for all

n, l i n z

and

(x_{k})_{k} s u b s e t E

,

E

a

n

dimensional subspace of

X

. In the case

q = 2

the same holds if we replace the expected value by the supremum.

Full work available at URL: https://arxiv.org/abs/math/9312206

zbMATH Keywords

eigenvalues approximation numbers absolutely summing operators weak cotype

Mathematics Subject Classification ID

Linear operators defined by compactness properties (47B07) Linear operators belonging to operator ideals (nuclear, (p)-summing, in the Schatten-von Neumann classes, etc.) (47B10) Riesz operators; eigenvalue distributions; approximation numbers, (s)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators (47B06) Probabilistic methods in Banach space theory (46B09)