Suppose that $\{X_k\}_{k\in\mathbb Z}$ is a weakly stationary sequence of random elements with values in a complex separable Hilbert space $\mathbb H$ and let us define the sequence of autocovariance operators by setting $$ C_t(h)=\operatorname E[\langle h,X_0\rangle X_t] $$ for each $t\in\mathbb Z$ and $h\in\mathbb H$. Are the autocovariance operators $C_t$ trace class for each $t\in\mathbb Z$? If they are, how can we prove that?
The autocovariance operators $C_t$ are not necessarily self-adjoint for $t\ne0$, nor are they necessarily non-negative definite for $t\ne0$. So it seems that we cannot determine if they are trace class by checking the convergence of the series $\sum_{n\ge1}\langle C_te_n,e_n\rangle$ for some orthonormal basis $\{e_n\}_{n\ge1}$ (see this question). However, we have that $\sum_{n\ge1}\langle C_te_n,e_n\rangle=\operatorname E\langle X_t,X_0\rangle$ and the right side of the equality is finite so the series on the left side converges.
Any help is much appreciated!
