Berezinians, exterior powers and recurrent sequences - To the memory of Felix Alexandrovich Berezin

We study power expansions of the characteristic function of a linear operator A in a p vertical bar 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 express the Berezinian of an operator as a ratio of two polynomial invariants. 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.