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Non-bounded linear operator \ell^2 \to \ell^2

Source: Miklos Schweitzer 2010 problem 7

November 14, 2017
college contestsMiklos Schweitzerreal analysis

Problem Statement

Is there any sequence (an)n=1∞(a_n)_{n=1}^{\infty} of non-negative numbers, for which ∑n=1∞an2<∞\sum_{n=1}^{\infty} a_n^2<\infty , but ∑n=1∞(∑k=1∞aknk)2=∞\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\frac{a_{kn}}{k} \right)^2=\infty ?
That contest - Miklos Schweitzer 2010- is missing on the contest page here for now being. The statements of all problems that year can be found [url=http://www.math.u-szeged.hu/~mmaroti/schweitzer/]here, but unfortunately only in Hungarian. I tried google translate but it was a mess. So, it would be wonderful if someone knows Hungarian and wish to translate it.
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