Topologically conjugate classification of diagonal operators
Yue Xin, Bingzhe Hou
TL;DR
The work addresses the topological classification of diagonal operators on the Banach space $\\ell^{p}$ up to topological conjugacy. It develops a framework of coordinatewise homeomorphisms $h^{\\mathfrak{S}}_{p}$ and scalar conjugacies $f_w$ to construct explicit conjugacies, with a key lemma establishing that $h^{\\mathfrak{S}}_{p}$ is a homeomorphism on $\\ell^{p}$. The results show that conjugacy classes are largely determined by the modulus sequence $|\\mathfrak{W}|$, yielding conjugacy to $2I$ or $\\tfrac{1}{2}I$ in certain spectral regimes, and proving non-conjugacy when any $|w_n|$ hits 1. This work clarifies when phase information is irrelevant to dynamics and provides concrete transformations between diagonal operators and canonical scalars, strengthening the link between operator theory and topological dynamics.
Abstract
Let $\ell^{p}$, $1\leq p<\infty$, be the Banach space of absolutely $p$-th power summable sequences and let $π_{n}$ be the natural projection to the $n$-th coordinate for $n\in\mathbb{N}$. Let $\mathfrak{W}=\{w_{n}\}_{n=1}^{\infty}$ be a bounded sequence of complex numbers. Define the operator $D_{\mathfrak{W}}: \ell^{p}\rightarrow\ell^{p}$ by, for any $x=(x_{1},x_{2},\ldots)\in \ell^p$, $π_{n}\circ D_{\mathfrak{W}}(x)=w_{n}x_{n}$ for all $n\geq1$. We call $D_{\mathfrak{W}}$ a diagonal operator on $\ell^{p}$. In this article, we study the topological conjugate classification of the diagonal operators on $\ell^{p}$. More precisely, we obtained the following results. $D_{\mathfrak{W}}$ and $D_{\vert\mathfrak{W}\vert}$ are topologically conjugate, where $\vert\mathfrak{W}\vert=\{\vert w_{n}\vert\}_{n=1}^{\infty}$. If $\inf_{n}\vert w_n\vert>1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $2\mathbf{I}$, where $\mathbf{I}$ means the identity operator. Similarly, if $\inf_{n}\vert w_n\vert>0$ and $\sup_{n}\vert w_n\vert<1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $\frac{1}{2}\mathbf{I}$. In addition, if $\inf_{n}\vert w_n\vert=1$ and $\inf_{n}\vert t_n\vert>1$, then $D_{\mathfrak{W}}$ and $D_{\mathfrak{T}}$ are not topologically conjugate.