# King's College London

## Sharp estimates for conditionally centred moments and for compact operators on $L^p$ spaces

Research output: Contribution to journalArticlepeer-review

Original language English 368-381 14 Mathematische Nachrichten 296 1 18 Dec 2022 15 Feb 2022 18 Dec 2022 29 Jan 2023 https://doi.org/10.1002/mana.202100217

## Abstract

Let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space, $\xi$ be a random variable on $(\Omega, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $\mathbf{E}^\mathcal{G} = \mathbf{ E}(\cdot | \mathcal{G})$ be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of $\xi - \mathbf{E}^\mathcal{G}\xi$ in terms of the moments of $\xi$. This allows us to find the optimal constant in the bounded compact approximation property of $L^p([0, 1])$, $1 < p < \infty$.

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