Invariant Subspaces and Quasiaffine Transforms of Unitary Operators

Abstract

A classical conjecture on nontrivial invariant subspaces for Hilbert-space contractions reads as follows. “a C ₁.-contraction has a nontrivial invariant subspace”. This turns out to be equivalent to a second conjecture, namely, “if a contraction is a quasiaffine transform of a unitary operator, then it has a nontrivial invariant subspace”. Although these are still unsolved problems, it can be proved that if a C ₁. -contraction has no nontrivial invariant subspace then it is a quasiaffine transform of its own unitary extension which is reductive and has an invariant dense and totally cyclic linear manifold. This paper presents a brief review, based on [7] and [9], on the equivalence between the above conjectures.