Given two arbitrary sequences
$({\lambda }_{j} )_{j\geq 1} $
and
$({\mu }_{j} )_{j\geq 1} $
of real numbers satisfying
$$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
we prove that there exists a unique sequence
$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $
, real valued, such that the Hankel operators
${\Gamma }_{c} $
and
${\Gamma }_{\tilde {c} } $
of symbols
$c= ({c}_{n} )_{n\geq 0} $
and
$\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $
, respectively, are selfadjoint compact operators on
${\ell }^{2} ({ \mathbb{Z} }_{+ } )$
and have the sequences
$({\lambda }_{j} )_{j\geq 1} $
and
$({\mu }_{j} )_{j\geq 1} $
, respectively, as non-zero eigenvalues. Moreover, we give an
explicit formula for
$c$
and we describe the kernel of
${\Gamma }_{c} $
and of
${\Gamma }_{\tilde {c} } $
in terms of the sequences
$({\lambda }_{j} )_{j\geq 1} $
and
$({\mu }_{j} )_{j\geq 1} $
. More generally, given two arbitrary sequences
$({\rho }_{j} )_{j\geq 1} $
and
$({\sigma }_{j} )_{j\geq 1} $
of positive numbers satisfying
$$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
we describe the set of sequences
$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $
of complex numbers such that the Hankel operators
${\Gamma }_{c} $
and
${\Gamma }_{\tilde {c} } $
are compact on
${\ell }^{2} ({ \mathbb{Z} }_{+ } )$
and have sequences
$({\rho }_{j} )_{j\geq 1} $
and
$({\sigma }_{j} )_{j\geq 1} $
, respectively, as non-zero singular values.