Quasiaffine orbits of invariant subspaces for uniform Jordan operators

TL;DR

This paper classifies invariant subspaces of uniform Jordan operators by characterizing when one subspace is in the quasiaffine orbit of another, based on quasisimilarity and compression injectability, refining prior results.

Contribution

It provides a precise criterion for the quasiaffine orbit relation of invariant subspaces for uniform Jordan operators, extending previous work by Bercovici and Smotzer.

Findings

Subspace $M_2$ is in the quasiaffine orbit of $M_1$ if and only if restrictions are quasisimilar.

The compression $T_{M_2^ot}$ can be injected into $T_{M_1^ot}$.

Refines previous classification results for invariant subspaces.

Abstract

We consider the problem of classification of invariant subspaces for the class of uniform Jordan operators. We show that given two invariant subspaces $M_1$ and $M_2$ of a uniform Jordan operator $T=S(\theta)\oplus S(\theta)\oplus \ldots$, the subspace $M_2$ belongs to the quasiaffine orbit of $M_1$ if and only if the restrictions $T|M_1$ and $T|M_2$ are quasisimilar and the compression $T_{M_2^\perp}$ can be injected in the compression $T_{M_1^\perp}$. Our result refines previous work on the subject by Bercovici and Smotzer.