# Matrix inequalities for the moment of the fixed Shatten norm

Let $$A_i, i=1, \ldots, N$$ be real (or complex) matrices of the same dimension. Let $$r_i, i=1, \ldots, N$$ be independent Rademacher random variables. The following inequality gives a bound on the expectation of the p-th Schatten norm: $$E\left\|\sum_{i=1}^N r_i A_i\right\|_{S_{p}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{p}}^{p}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{p}}^{p}\right\}.$$ Question: whether some similar bounds known for p-th moment of the fixed q-th Schatten norm, i.e. something like $$E\left\|\sum_{i=1}^N r_i A_i\right\|_{S_{q}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{q}}^{p}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{q}}^{p}\right\}.$$

If you do not care about the precise constants, the answer is yes, and this follows directely from the Kahane inequality (which says that, in any normed space and $$1 \leq p,q<\infty$$, the $$L_p$$-norm of $$\sum r_i x_i$$ is dominated by its $$L_q$$-norm, up to constant depending on $$p,q$$ only). The conclusion is that for every $$1 \leq p, q <\infty$$, there is a constant $$C_{p,q}$$ such that $$\left(E \|\sum_{i=1}^N r_i A_i\|_{S_{q}}^{p}\right)^{\frac 1 p}\leq C_{p,q} \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{q}}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{q}}\right\}.$$
Observe that it is necessary to take the $$1/p$$-th power if you want to have $$C_{p,p} \leq C \sqrt{p}$$.
If you care about the constants, what you ask for is way too strong. It is clearly necessary to have $$C_{p,q} \to \infty$$ as $$q \to \infty$$ (as otherwise by considering diagonal matrices it would give Khintchine inequalities in $$\ell_\infty$$).