  Analogs of the classical Sylvester theorem have been known for matrices with
entries in noncommutative algebras including the quantized algebra of functions
on GL(N) and the Yangian for gl(N). We prove a version of this theorem for the
twisted Yangians Y(g(N)) associated with the orthogonal and symplectic Lie
algebras g(N)=o(N) or sp(N). This gives rise to representations of the twisted
Yangian Y(g(N-M)) on the space of g(M)-homomorphisms Hom(W,V), where W and V
are finite-dimensional irreducible modules over g(M) and g(N), respectively. In
the symplectic case these representations turn out to be irreducible and we
identify them by calculating the corresponding Drinfeld polynomials. We also
apply the quantum Sylvester theorem to realize the twisted Yangian as a
projective limit of certain centralizers in universal enveloping algebras.
