Consequences of universality among Toeplitz operators
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.
scientific article; zbMATH DE number 6479724
scientific article