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On the Hyperinvariant Subspace Problem. IV

Published online by Cambridge University Press:  20 November 2018

H. Bercovici ,
C. Foias  and
C. Pearcy
H. Bercovici
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. e-mail:[email protected]
C. Foias
Affiliation:
Department of Mathematics Texas A&M University College Station, TX 77843 U.S.A. e-mail:[email protected]
C. Pearcy
Affiliation:
Department of Mathematics Texas A&M University College Station, TX 77843 U.S.A. e-mail:[email protected]
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Abstract

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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, ${{C}_{00}}$, $(\text{BCP})$-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.

Keywords

47A15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 4 , 01 August 2008 , pp. 758 - 789
Copyright
Copyright © Canadian Mathematical Society 2008

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