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Let $M_n : H_1 \to H_2$ be bounded linear maps from two Hilbert spaces such that$\sum_{n} M_n^*M_n$ is strongly convergent to the identity operator of $H_1$. Then, what can I say about $\sum_{n} M_n M_n^*$? Does it converge to some bounded operator on $H_2$? I cannot figure out anything about $\sum_{n} M_n M_n^*$ at all...

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No, you cannot say much. Consider these two examples:

  1. $\{M_n\}$ is a sequence of rank-one projections that adds to the identity. Then $$ \sum_mM_n^*M_n=\sum_nM_nM_n^*=I. $$

  2. Decompose $H$ as a direct sum of infinitely many infinite-dimensional subspaces $H_n$. Let $X_n$ be the obvious isometry $X_n:H\to H_n$. Then $X_n^*X_n=I$ and $X_nX_n^*$ is the orthogonal projection onto $H_n$. Take $M_n=X_n^*$. Then $$ \sum_nM_n^*M_n=\sum_nX_nX_n^*=I, $$ while $\sum_n M_nM_n^*$ does not exist as $M_nM_n^*=X_n^*X_n=I$ for all $n$.

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