They presented their result last friday at Congreso Bianual de la Real Sociedad Española de Matemáticas. Carl Cowen gave a talk entitled “Rota’s Universal Operators and invariant subspaces in Hilbert spaces.”
I still don’t have access to the preprint but I think that they use the following reduction of Caradus [Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23 (1969) 526–527]: it suffices to show that the minimal invariant subspaces of a universal operator are all finite dimensional.
A bounded linear operator on a Hilbert space is said to be universal provided that for any bounded linear operator on there is a closed subspace invariant under and there is a nonzero scalar such that is similar to
Caradus showed that if is surjective and is infinite dimensional then is universal. The first example of a universal operator was discovered by G.C. Rota, [On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469-472]: a backward shift of infinite multiplicity.
Update 04 February 2013: I’ve been told that Donald Sarason from University of California at Berkeley has found a gap in the proof.