Let
- $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space
- $U,H$ be separable $\mathbb R$-Hilbert spaces
- $(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$ be orthonormal bases of $U$ and $H$, respectively
- $M$ and $N$ be $U$- and $H$-valued square-integrable continuous martingales on $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$, respectively
Note that $$[\![M,N]\!]_t:=\sum_{(m,\:n)\in\mathbb N^2}[M^m,N^n]_te_m\otimes f_n\;\;\;\text{for }t\ge0$$ takes values in $\mathfrak L_1(U,H)$ (trace class operators) and is of bounded variation. Moreover, $M\otimes N-[\![M,N]\!]$ is a martingale. Analogously, let $[\![M]\!]:=\sum_{(m,\:n)\in\mathbb N^2}[M^m,M^n]_te_m\otimes e_n$ and $[\![N]\!]:=\sum_{(m,\:n)\in\mathbb N^2}[N^m,N^n]_tf_m\otimes f_n$ and note that $[\![M]\!]$ and $[\![N]\!]$ take values in $\mathfrak L_1^+(U)$ (nonnegative and self-adjoint trace class operators) and $\mathfrak L_1^+(H)$, respectively.
We can show that the Lebesgue-Stieltjes measure ${\rm d}[\![M,N]\!]$ is absolutely continuous with respect to ${\rm d}([M]+[N])$ ($[M]$ and $[N]$ denoting the quadratic variation of $M$ and $N$, respectively. Hence, $$[\![M,N]\!]_t=\int_{(0,\:t]}\operatorname{Cor}(M,N)\:{\rm d}([M]+[N])\tag1$$ for some $\mathfrak L_1(U,H)$-valued process $\operatorname{Cor}(M,N)$. Analogously, define $\operatorname{Cov}(M):=\frac12\operatorname{Cor}(M,M)$ such that $$[\![M]\!]_t=\int_{(0,\:t]}\operatorname{Cov}(M)\:{\rm d}[M]\tag2.$$ Define $\operatorname{Cov}(N)$ in the same way.
Let $\tilde U$ and $\tilde H$ be separable $\mathbb R$-Hilbert spaces. If $\Phi\in\operatorname{HS}(\operatorname{Cov}(M)^{1/2}U,\tilde U)$ (Hilbert-Schmidt operators) and $\Psi\in\operatorname{HS}(\operatorname{Cov}(N)^{1/2}H,\tilde H)$, are we able to show that $\Psi\operatorname{Cor}(M,N)\Phi^\ast$ is well-defined and trace-class?
