Abstract. It is shown that a class of composition operators $C_\phi $ has the property that for every $\lambda $ in the interior of the spectrum of $C_\phi $ the operator $U=C_\phi -\lambda{\rm Id}$ is universal in the sense of Caradus, i.e., every Hilbert space operator has a non-zero multiple similar to the restriction of $U$ to an invariant subspace. As a generalization, weighted composition operators on the $L^2$ and Sobolev spaces of the unit interval are shown to have the same property and thus a complete knowledge of their minimal invariant subspaces would imply a solution to the invariant subspace problem for Hilbert space. Moreover, a generalization of sufficient conditions for an operator to be universal is obtained. Cyclicity and non-cyclicity results for a certain class of weights and composition functions are also proved.

AMS Subject Classification (1991): 47A15, 47B33, 47A16

Keyword(s): weighted composition operators, semi-Fredholm operators, universal operators, invariant subspaces, cyclic vectors, Müntz theorem

Received October 9, 2011, and in final form March 30, 2012. (Registered under 52/2011.)