If $U_0$ and $H$ are separable $\mathbb R$-Hilbert spaces, then a bounded linear operator $\Phi:U_0\to H$ is called Hilbert-Schmidt $:\Leftrightarrow$ $$\left\|\Phi\right\|_{\operatorname{HS}(U_0,\:H)}^2:=\sum_{n\in\mathbb N}\left\|\Phi e_0^n\right\|_H^2<\infty\tag 1$$ for any orthonormal basis $(e_0^n)_{n\in\mathbb N}$ of $U_0$. $(1)$ is well-defined, cause we can show that its value doesn't depend on the choice of $(e_0^n)_{n\in\mathbb N}$.
Let $\mathfrak L(A,B)$ denote the space of bounded linear operators from $A$ to $B$ and suppose $\Psi\in\mathfrak L(U_0,\mathfrak L(U_0,H))$. I've found the symbol sequence $\left\|\Psi\right\|_{\operatorname{HS}^{(2)}(U_0,\:H)}$ in some papers I've read, e.g. in A Runge–Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise in equation (2.6), without a definition.
I suppose the notion of being Hilbert-Schmidt can be defined for multilinear bounded operators too, but it's not clear to me how exactly $\left\|\Psi\right\|_{\operatorname{HS}^{(2)}(U_0,\:H)}$ is defined.
One option would be $$\left\|\Psi\right\|_{\operatorname{HS}^{(2)}(U_0,\:H)}^2=\sum_{n\in\mathbb N}\left\|(\Psi e^n_0)e_0^n\right\|_H^2\;.\tag 2$$ Especially for $Q\in\mathfrak L(U_0,\mathfrak L(V_0,H))$, where $V_0$ is another separable $\mathbb R$-Hilbert space, another reasonable option would be $$\left\|Q\right\|_{\operatorname{HS}^{(2)}}^2:=\sum_{m\in\mathbb N}\sum_{n\in\mathbb N}\left\|(Q e_0^m)f_0^n\right\|_H^2\tag 3\;,$$ where $(f_0^n)_{n\in\mathbb N}$ is an orthonormal basis of $V_0$.
So, what's the actual definition of $\left\|\Psi\right\|_{\operatorname{HS}^{(2)}(U_0,\:H)}$ and $\left\|Q\right\|_{\operatorname{HS}^{(2)}}$?