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In general, a Hilbert Schmidt Operator $M$ is one where \begin{equation} \sum_{i=1}^{\infty} ||M \varphi_{i}||^2 \end{equation} is bounded for any orthonormal system $\{\varphi_i: i\in \mathbb{N}\}$.
Now, let $M_k$ be a sequence of Hilbert Schmidt operators. We call it a uniform sequence of Hilbert Schmidt operators, if for every $\rho>0$ there is an $n \in \mathbb{N}$ such that \begin{equation} \sum_{i=n}^{\infty} ||M_k \varphi_{i}||^2 \leq \rho \end{equation} for all $k \in \mathbb{N}$.
Further remarks:
If the observation operator $H$ is depending on $k$, i.e. $H = H_k$, we need to have a sequence $\lambda_{j}^{\ast} > 0$ such that for the singular values $\lambda_{j}^{(k)}$ of $H_k$ we have \begin{equation} \lambda_{j}^{(k)} \geq \lambda_{j}^{\ast} \end{equation} for all $j, k \in \mathbb{N}$.
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