Sharp estimates for conditionally centered moments and for compact operators on Lp$L∧p$ spaces

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