Optimal Constants in Khintchine Type Inequalities for Fermions, Rademachers and $q$-Gaussian Operators
Volume 53 / 2005
Bulletin Polish Acad. Sci. Math. 53 (2005), 315-321
MSC: Primary 46L53; Secondary 46L52.
DOI: 10.4064/ba53-3-9
Abstract
For $(P_{k})$ being Rademacher, Fermion or $q$-Gaussian ($-1\leq q\leq0$) operators, we find the optimal constants $C_{2n}$, $n\in\Bbb N$, in the inequality $$ \Big\| \sum_{k=1}^N A_k \otimes P_k \Big\|_{2n} \leq [C_{2n}]^{1/2n} \max \Big\{ \Big\| \Big( \sum_{k=1}^N A_k^* A_k \Big)^{1/2} \Big\|_{L_{2n}}, \Big\| \Big( \sum_{k=1}^N A_k A_k^* \Big)^{1/2} \Big\|_{L_{2n}} \Big\},$$ valid for all finite sequences of operators $(A_{k})$ in the non-commutative $L_{2n}$ space related to a semifinite von Neumann algebra with trace. In particular, $C_{2n}=(2nr-1)!!$ for the Rademacher and Fermion sequences.
