Journal of Operator Theory
Volume 94, Issue 2, Autumn 2025 pp. 445-455.
Noncommutative sharp dual Doob inequalitiesAuthors: Fedor Sukochev (1), Dejian Zhou (2)
Author institution: (1) School of Mathematics and Statistics, University Of New South Wales, Kensington, 2033, Australia
(2) School of Mathematics and Statistics, Central South University, Changsha, 410083, China
Summary: Let $(x_k)_{k=1}^n$ be a sequence of positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras $(\mathcal{M}_n)_{k\geqslant 1}$ of the finite von Neumann algebra $\mathcal{M}$. We establish the following sharp noncommutative dual Doob inequalities: $\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})}\leqslant \frac{1}{p} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})},\quad p\in(0, 1],$ and $\Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})}\leqslant p\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})},\quad p\in[1,2].$ As applications, we obtain several noncommutative martingale inequalities with better constants.
DOI: http://dx.doi.org/10.7900/jot.2024jan28.2457
Keywords: noncommutative martingales, dual Doob inequality, best constants
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