On Hyperinvariant Subspaces of Operators Containing Unilateral Shifts

Abstract

Abstract The hyperinvariant subspace problem for Hilbert space operators T containing a unilateral shift is addressed. The discussion is based on a similarity model of T , which is an operator-matrix $$\widehat{T}= [T_{i,j}]_3$$ T ^ = [ T i , j ] 3 where $$T_{1,1}$$ T 1 , 1 is the simple unilateral shift S and $$T_{3,3}$$ T 3 , 3 is a cyclic diagonal operator D . The existence of D is established by the technique resulting almost invariant half-spaces in [1]; see also [10, 24]. For any operator $$Q=[Q_{i,j}]_3$$ Q = [ Q i , j ] 3 in the commutant of $$\widehat{T}$$ T ^ , the entry $$Q_{3,1}$$ Q 3 , 1 intertwines S with D up to a transformation of rank at most 1. These entries form a linear manifold $$\mathcal{L}_{3,1}$$ L 3 , 1 . We focus on 3-dimensional cross-sections of $$\mathcal{L}_{3,1}$$ L 3 , 1 . These are subspaces of $$3\times 3$$ 3 × 3 complex matrices, transformed into singular matrices by a canonical mapping. If such a subspace $$\mathcal{L}$$ L is not transitive, then T has a nontrivial hyperinvariant subspace. A throrough study reveals that $$\mathcal{L}$$ L can be transitive only if it has a very specific basis. Consequences of the existence of nontrivial hyperinvariant subspaces in the presence of shift-type invariant subspaces are also discussed.