On Hyperinvariant Subspaces of Operators Containing Unilateral Shifts

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 T= [Ti,j]₃ T ^ = [ T i , j ] 3 where T1,1 T 1 , 1 is the simple unilateral shift S and T3,3 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=[Qi,j]₃ Q = [ Q i , j ] 3 in the commutant of T T ^ , the entry Q3,1 Q 3 , 1 intertwines S with D up to a transformation of rank at most 1. These entries form a linear manifold L3,1 L 3 , 1 . We focus on 3-dimensional cross-sections of L3,1 L 3 , 1 . These are subspaces of 3× 3 3 × 3 complex matrices, transformed into singular matrices by a canonical mapping. If such a subspace L L is not transitive, then T has a nontrivial hyperinvariant subspace. A throrough study reveals that 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.