Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-03T00:55:45.971Z Has data issue: false hasContentIssue false

Invariant subspaces for spherical contractions

Published online by Cambridge University Press:  01 July 1997

J Eschmeier
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK Present address: Fachbereich Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany. Email: [email protected]
Get access

Abstract

Let $T$ be a contraction on a complex Hilbert space $H$. A result of Brown, Chevreau and Pearcy from 1979 shows that $T$ has a non-trivial invariant subspace if the spectrum of $T$ is dominating in the open unit disc. It is the purpose of the present paper to prove the multidimensional analogue of this result for spherical contractions $T \in L(H)^n$ that possess a spherical dilation and for which the Harte spectrum is dominating in the open unit ball $B$ in $\mathbb{C}^n$. If even the essential Harte spectrum of $T$ is dominating in $B$, then $T$ is shown to be reflexive and to possess an extremely rich invariant subspace lattice. The proof is based on the existence of an $H^{\infty}$-functional calculus for completely non-unitary spherical contractions and on a multidimensional analogue of the classical result of Sz.Nagy and Foias, stating that each spherical contraction which is neither of type $C_{\cdot 0}$ nor of type $C_{0 \cdot}$ and which does not consist of multiples of the identity operator on $H$, possesses non-trivial joint hyperinvariant subspaces.

1991 Mathematics Subject Classification: 47A13, 47A15, 47A60

Type
Research Article
Copyright
London Mathematical Society 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)