For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$.


It is well-know that a series $\sum\_{n\in\omega}r\_n$ of real numbers converges unconditionally if and only if $\sum\_{n\in\omega}|r\_n|<\infty$.


On the other hand, there exists an unconditionally convergent series $\sum\_{n\in\omega}x\_n$ in $\ell\_2$ such that the series $\sum\_{n\in\omega}|x\_n|$ is divergent.


I am interested in finding conditions on an unconditionally convergent series $\sum\_{n\in\omega}x\_n$ in $\ell\_2$ guaranteeing that the series $\sum\_{n\in\omega}|x\_n|$ converges.



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> **Question.** Let $I:\ell\_1\to\ell\_2$ be the identity operator and $(x\_n)\_{n\in\omega}$ be a sequence of elements of $\ell\_1$ such that $\sum\_{n\in\omega}\|x\_n\|^2<\infty$ and the series $\sum\_{n\in\omega}I(x\_n)$ unconditionally converges in $\ell\_2$. Is the series $\sum\_{n\in\omega}|I(x\_n)|$ convergent in $\ell\_2$?
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Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors
If I understand correctly what you are asking, the answer is "certainly not".


Consider $k$ orthogonal vectors $v\_j$ with $k$ coordinates $\pm 1$ (the Hadamard matrix). Now multiply them by $t$ and take $n$ such identical bunches. Then the sum of $\ell^1$ norms squared is $nt^2k^3$, the squared $\ell^2$ norm of the sum of absolute value vectors is $n^2t^2k^3$ and the maximum of the squared $\ell^2$ norms of subset sums is $n^2t^2k^2$. Then playing this game on disjoint bunches of coordinates, we see that the question is whether the conditions $\sum\_m n\_mt\_m^2k\_m^3<+\infty$ and $\sum\_m n\_m^2t\_m^2k\_m^2<+\infty$ imply $\sum\_m n\_m^2t\_m^2k\_m^3<+\infty$. Now just take $n\_m=k\_m=2^m$ and $t\_m=2^{-2m}/m$, say.


But perhaps you meant something else? :-)


