Let the linear operator $T:l^2\rightarrow l^2$ be defined by $y=Tx$ where $x=\{\xi_j\}$, $y=\{\eta_j\}$, and $\eta_j = \alpha_j \xi_j$, where $\{\alpha_j\}$ is a dense sequence in $[0,1]$. Does $y=Tx\in l^2, \forall x\in l^2$, imply that $\{\alpha_j\}\in l^2$?
In Kreyszig's "Introductory Functional Analysis with Applications", he defines such an operator in Sec. 7.3, Problem 4 and asks about the spectrum.