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 $\widehat T= [T_{i,j}]_3$ where $T_{1,1}$ is the simple unilateral shift $S$ and $T_{3,3}$ is a cyclic diagonal operator $D$. The existence of $D$ is established by the technique resulting almost invariant half-spaces in \cite{APTT}; see also \cite{Tc} and \cite{HP}. For any operator $Q=[Q_{i,j}]_3$ in the commutant of $\widehat T$, the entry $Q_{3,1}$ intertwines $S$ with $D$ up to a transformation of rank at most 1. These entries form a linear manifold ${\cal L}_{3,1}$. We focus on 3-dimensional cross-sections of ${\cal L}_{3,1}$. These are subspaces of $3\times 3$ complex matrices, transformed into singular matrices by a canonical mapping. If such a subspace ${\cal L}$ is not transitive, then $T$ has a nontrivial hyperinvariant subspace. A throrough study reveals that ${\cal L}$ 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.