# Inverse Problems and Data Assimilation Wiki and International Community

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# Playing

In general, a Hilbert Schmidt Operator $M$ is one where $$\sum_{i=1}^{\infty} ||M \varphi_{i}||^2$$ 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 $$\sum_{i=n}^{\infty} ||M_k \varphi_{i}||^2 \leq \rho$$ 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 $$\lambda_{j}^{(k)} \geq \lambda_{j}^{\ast}$$ for all $j, k \in \mathbb{N}$.