  We study power expansions of the characteristic function of a linear operator
$A$ in a $p|q$-dimensional superspace $V$. We show that traces of exterior
powers of $A$ satisfy universal recurrence relations of period $q$.
`Underlying' recurrence relations hold in the Grothendieck ring of
representations of $\GL(V)$. They are expressed by vanishing of certain Hankel
determinants of order $q+1$ in this ring, which generalizes the vanishing of
sufficiently high exterior powers of an ordinary vector space. In particular,
this allows to explicitly express the Berezinian of an operator as a rational
function of traces. We analyze the Cayley--Hamilton identity in a superspace.
Using the geometric meaning of the Berezinian we also give a simple formulation
of the analog of Cramer's rule.
