  A depending on a complex parameter $k$ superanalog ${\mathcal S}{\mathcal L}$
of Calogero operator is constructed; it is related with the root system of the
Lie superalgebra ${\mathfrak{gl}}(n|m)$. For $m=0$ we obtain the usual Calogero
operator; for $m=1$ we obtain, up to a change of indeterminates and parameter
$k$ the operator constructed by Veselov, Chalykh and Feigin [2,3]. For $k=1,
\frac12$ the operator ${\mathcal S}{\mathcal L}$ is the radial part of the 2nd
order Laplace operator for the symmetric superspaces corresponding to pairs
$(GL(V)\times GL(V), GL(V))$ and $(GL(V), OSp(V))$, respectively. We will show
that for the generic $m$ and $n$ the superanalogs of the Jack polynomials
constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of
${\mathcal S}{\mathcal L}$; for $k=1, \frac12$ they coinside with the spherical
functions corresponding to the above mentioned symmetric superspaces. We also
study the inner product induced by Berezin's integral on these superspaces.
