Abstract view
On the Hyperinvariant Subspace Problem. IV


Published:20080801
Printed: Aug 2008
H. Bercovici
C. Foias
C. Pearcy
Abstract
This paper is a continuation of three recent articles
concerning the structure of hyperinvariant
subspace lattices of operators on a (separable, infinite dimensional)
Hilbert space $\mathcal{H}$. We show herein, in particular, that
there exists a ``universal'' fixed blockdiagonal operator $B$ on
$\mathcal{H}$ such that if $\varepsilon>0$ is given and $T$ is
an arbitrary nonalgebraic operator on $\mathcal{H}$, then there exists
a compact operator $K$ of norm less than $\varepsilon$ such that
(i) $\Hlat(T)$ is isomorphic as a complete lattice to $\Hlat(B+K)$
and (ii) $B+K$ is a quasidiagonal, $C_{00}$, (BCP)operator with
spectrum and left essential spectrum the unit disc. In the last four
sections of the paper, we investigate the possible structures of the
hyperlattice of an arbitrary algebraic operator. Contrary to existing
conjectures, $\Hlat(T)$ need not be generated by the ranges and kernels
of the powers of $T$ in the nilpotent case. In fact, this lattice
can be infinite.