Topologically conjugate classification of diagonal operators

Abstract

Let ${\ell }^p$, $1\le p<\infty$, be the Banach space of absolutely $p$-th power summable sequences and let $\ddot{I}{€}_n$ 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 $D_{\mathfrak{W}}:{\ell }^p\to {\ell }^p$ by, for any $x=(x_1,x_2,\dots )\in {\ell }^p$, $\ddot{I}{€}_n\circ D_{\mathfrak{W}}(x)=w_n{x}_n$ for all $n\ge 1$. 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 $D_{|\mathfrak{W}|}$ are topologically conjugate, where $|\mathfrak{W}|=\{|w_n|{\}}_{n=1}^{\infty }$. If ${\inf }_n|w_n|>1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $2\mathbf{I}$, where $\mathbf{I}$ means the identity operator. Similarly, if ${\inf }_n|w_n|>0$ and ${\sup }_n|w_n|<1$, then $D_{\mathfrak{W}}$ is topologically conjugate to $\frac{1}{2}\mathbf{I}$. In addition, if ${\inf }_n|w_n|=1$ and ${\inf }_n|t_n|>1$, then $D_{\mathfrak{W}}$ and $D_{\mathfrak{T}}$ are not topologically conjugate.

16 pages