Consequences of universality among Toeplitz operators
type: Item
instance of: scholarly article
title: Consequences of universality among Toeplitz operators
zbMATH Open document ID: 1334.47011
author: Carl C. Cowen
author: Eva A. Gallardo Gutiérrez
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-09-09T00:00:00Z
review text: Let \(\mathcal H\) be a separable infinite dimensional Hilbert space. An operator \(U \in \mathcal B(\mathcal H)\) is called universal if for each non-zero operator \(A \in \mathcal B(\mathcal H)\) there exist a \(U\)-invariant subspace \(M\subset \mathcal H\) and a non-zero \(\lambda\in\mathbb C\) such that \(\lambda A\) is similar to \(U|_{M}\). In the paper [Commun. Pure Appl. Math. 13, 469--472 (1960; Zbl 0097.31604)], \textit{G.-C. Rota} proved the existence of such operators. The authors of the paper under review prove the following generalization of the result obtained in [\textit{S. R. Caradus}, Proc. Am. Math. Soc. 23, 526--527 (1969; Zbl 0186.19204)]: If \(U\) is such an operator in \(\mathcal B(\mathcal H)\) that \(\dim\ker U=\infty\), \(\text{ran} U=\mathcal H\), then there exists \(\varepsilon>0\) so that for all \(\mu\), \(|\mu| < \varepsilon\) the operator \(U+\mu I\) is universal. The paper contains examples of Toeplitz operators on the standard Hardy space whose adjoints are universal operators. Certain properties of these operators are investigated. Some other results in the domain under consideration are obtained. Regarding the invariant subspace problem, the authors pose three questions connecting shift invariant subspaces and invariant subspaces of analytic Toeplitz operators.
reviewed by: Vladimir S. Pilidi
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 47B32
zbMATH DE Number: 6479724
zbMATH Keywords: invariant subspace
zbMATH Keywords: Hardy space
zbMATH Keywords: composition operator
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: weighted composition operator
zbMATH Keywords: Rota's universal operators
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.06.061
OpenAlex ID: W1700886700
cites work: On functions orthogonal to invariant subspaces
cites work: Universal Operators and Invariant Subspaces
cites work: Q3091640
cites work: The Commutant of an Analytic Toeplitz Operator
cites work: Q4195461
cites work: An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators
cites work: Unitary equivalence of one-parameter groups of Toeplitz and composition operators
cites work: Q4888816
cites work: Q5660202
cites work: Cyclic vectors and invariant subspaces of the backward shift operator
cites work: Minimal invariant subspaces for composition operators
cites work: Compact weighted composition operators on the Hardy space
cites work: On Analytically Invariant Subspaces and Spectra
cites work: Noncyclic Vectors for S ∗
cites work: Operators, functions, and systems: an easy reading. Volume I: Hardy, Hankel, and Toeplitz. Transl. from the French by Andreas Hartmann
cites work: Q3733578
cites work: Invertible composition operators on \(H^ p\)
cites work: Q5686535
cites work: On models for linear operators
DOI: 10.1016/J.JMAA.2015.06.061
description: scientific article; zbMATH DE number 6479724