Norm closures of orbits of bounded operators

TL;DR

This paper characterizes the norm closures of operator orbits in Hilbert spaces using specific cardinal invariants and binary parameters, providing a detailed classification based on group actions.

Contribution

It introduces a new framework using cardinal invariants and binary parameters to describe the norm closures of operator orbits under various group actions.

Findings

Provides explicit descriptions of orbit closures for different group choices.

Introduces three cardinal invariants and a binary parameter to classify operators.

Offers a comprehensive classification scheme for operator orbits.

Abstract

To every bounded linear operator $A$ between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ three cardinals $\iota_r(A)$, $\iota_i(A)$ and $\iota_f(A)$ and a binary number $\iota_b(A)$ are assigned in terms of which the descriptions of the norm closures of the orbits $\{G A L^{-1}:\ L \in \mathcal{G}_1,\ G \in \mathcal{G}_2\}$ are given for $\mathcal{G}_1$ and $\mathcal{G}_2$ (chosen independently) being the trivial group, the unitary group or the group of all invertible operators on $\mathcal{H}$ and $\mathcal{K}$, respectively.