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# On the Hyperinvariant Subspace Problem. IV

Published:2008-08-01
Printed: Aug 2008
• H. Bercovici
• C. Foias
• C. Pearcy
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## 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 block-diagonal 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.