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While doing my bachelor's thesis, I got stuck on a problem which basically comes down to the following:

Suppose $\mathcal{H}$ is a Hilber space and let the (not necessarily finite-dimensional) symmetric matrix $A=(a_{ij})$ be a bounded operator on this Hilbert space. Let $\epsilon > 0$ and suppose $\sum_j{a_{ij}} \leq \epsilon$, for all $i$. This implies that the operator norm $||A|| \leq \epsilon$.

I don't really see how to do this. My professor (who is absent the next couple of days) said that this for example can be proven by "Schur's criteria", but I've been googling now for a while and I still haven't found a criteria (by Schur) which can be useful. Can someone give me a reference, or an other insight into the problem (not necessarily related to Schur)?

Martin Argerami
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  • This is what he means, but unfortunately is stated in terms of integration instead of a Schoeder basis. The whole thing translates over to your case with $T \equiv A$, $K(x,y) \equiv (a_{ij})$ and integration replaced by summation. – Paul Sinclair Mar 10 '18 at 00:40
  • @PaulSinclair Thanks for the answer! I found this: https://math.stackexchange.com/questions/1243034/prove-that-an-infinite-matrix-defines-a-compact-operator-on-l2, which actually gives a straight answer to my question. In my case $p_i=q_j=1$ and $\beta=\gamma=\epsilon$. I forgot to mention that the Hilbert space $\mathcal{H}$ I'm working with is a $l²$-space... –  Mar 10 '18 at 09:29

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