  A linear operator $U$ acting boundedly on an infinite-dimensional separable
complex Hilbert space $H$ is universal if every linear bounded operator acting
on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its
invariant subspaces. It turns out that characterizing the lattice of closed
invariant subspaces of a universal operator is equivalent to solve the
Invariant Subspace Problem for Hilbert spaces. In this paper, we consider
invertible weighted hyperbolic composition operators and we prove the
universality of the translations by eigenvalues of such operators, acting on
Hardy and weighted Bergman spaces. Some consequences for the Banach space case
are also discussed.
