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Let $\{A_{j}\}$ be a sequence of bounded operators on a Hilbert space satisfying $\|A_{j}^{\ast}A_{k}\| \leq C_{j - k}$ and $\|A_{j}A_{k}^{\ast}\| \leq C_{j - k}$ where $\sum C_{i} < \infty$. Fix an $x$ in our Hilbert space. Why does $$\sum_{i = 1}^{\infty}\sum_{j = 1}^{\infty}|\langle A_{i}x, A_{j}x \rangle|$$ converge?

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I believe you meant "$C_{j-k}^2$" where it reads "$C_{j-k}$" and in this case the result follows from the proof of Cotlar's Lemma (see http://en.wikipedia.org/wiki/Cotlar%E2%80%93Stein_lemma).

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