  For each sequence $\{c_n\}_n$ in $l_{1}(\N)$ we define an operator $A$ in the
hyperfinite $\mathrm{II}_1$-factor $\mathcal{R}$. We prove that these operators
are quasinilpotent and they generate the whole hyperfinite
$\mathrm{II}_1$-factor. We show that they have non-trivial, closed, invariant
subspaces affiliated to the von Neumann algebra and we provide enough evidence
to suggest that these operators are interesting for the hyperinvariant subspace
problem. We also present some of their properties. In particular, we show that
the real and imaginary part of $A$ are equally distributed, and we find a
combinatorial formula as well as an analytical way to compute their moments. We
present a combinatorial way of computing the moments of $A^{*}A$.
