The invariant subspace problem

It appears that Carl Cowen from Purdue University and Eva Gallardo from Universidad Complutense de Madrid have found an affirmative solution to the invariant subspace problem on Hilbert spaces.

evagallardoycarlcowen01

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 U on a Hilbert space H is said to be universal provided that for any bounded linear operator T on H there is a closed subspace H_0 \subseteq H invariant under U and there is a nonzero scalar \lambda \in \mathbb{C} such that \lambda T is similar to U_{|H_0}.

Caradus showed that if U is surjective and \ker U is infinite dimensional then U 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.

Acerca de Miguel Lacruz

Gijón, Asturias, España, 1963
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