Journal of Operator Theory

Volume 93, Issue 2, Spring 2025 pp. 355-391.

On expansive operators that are quasisimilar to the unilateral shift of finite multiplicity

Authors: Maria F. Gamal'
Author institution: St. Petersburg Branch, V.A. Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023, Russia

Summary: An operator $T$ on a Hilbert space $\mathcal H$ is called expansive, if $\|Tx\|\geqslant \|x\|$ ($x\in\mathcal H$). Expansive operators $T$ quasisimilar to the unilateral shift of finite multiplicity $N$ are studied. In particular, it is proved that $I-T^*T$ is of trace class for such $T$ and there exist invariant subspaces $\mathcal M$ of $T$ such that the restriction $T|_{\mathcal M}$ of $T$ on $\mathcal M$ is similar to the unilateral shift of multiplicity $1$. Such $\mathcal M$ span $\mathcal H$, and the minimal number of such $\mathcal M$ which span $\mathcal H$ is $N$ for $N\geqslant 2$ and $2$ for $N=1$.

DOI: http://dx.doi.org/10.7900/jot.2023apr27.2427
Keywords: expansive operator, contraction, quasisimilarity, similarity, unilateral shift, invariant subspaces, unitary asymptote, intertwining relation

Contents Full-Text PDF