The problem I have comes from Walter Rudin's Functional Analysis, chapter 10 exercise 19. The exercise begins with the following:

Let $S_R$ be the right shift operator, acting on $\ell^2=\ell^2(\mathbb{N})$. Let $\{c_n\}$ be a sequence of complex numbers such that $c_n\neq0$ but $c_n\to0$ as $n\to\infty$.
Define $M\in\mathcal{B}(\ell^2)$ by \begin{align*} (Mf)(n)=c_nf(n)\qquad(n\in\mathbb{N}), \end{align*} and define $T\in\mathcal{B}(\ell^2)$ by $T=MS_R$.

I need to show that the spectrum $\sigma(T)$ of $T$ is $\{0\}$. So far I have shown that the operator norm of $T^m$ is $b_m=\sup\{|c_{n-m+1}\cdots c_n|:n>m\}$. Since the spectral radius $\rho(T)$ is the limit of $\|T^m\|^{1/m}$, as $m\to\infty$, I will be done if I can show that $(b_m)^{1/m}\to0$ as $m\to\infty$. However, I am having trouble showing this.

Any hints as to how I should proceed would be greatly appreciated.


Notice that, the way you have defined $T=MS_R$, we have \begin{equation}T^1(a_1,a_2,a_3,\cdots)=(0,c_2a_1,c_3a_2,c_4a_3,\cdots),\end{equation} \begin{equation}T^2(a_1,a_2,a_3,\cdots)=(0,0,c_3c_2a_1,c_4c_3a_2,c_5c_4a_3,\cdots),\end{equation} and so on. In particular, if $(e_i)_{i=1}^\infty$ is the canonical basis for $\ell_p$, $p=2$, then we have \begin{equation}T^m\sum_{i=1}^\infty a_ie_i=\sum_{i=1}^\infty \left(\sum_{j=1+i}^{m+i}c_j\right)a_ie_{m+i}\end{equation} and hence, by Holder and symmetry of $(e_i)_{i=1}^\infty$, \begin{equation}\|T^m\sum_{i=1}^\infty a_ie_i\|=\|\sum_{i=1}^\infty \left(\prod_{j=1+i}^{m+i}c_j\right)a_ie_{m+i}\|\leq\sup_{i\in\mathbb{N}}|\prod_{j=1+i}^{m+i}c_j|\|\sum_{i=1}^\infty a_ie_i\|\end{equation} Thus, \begin{equation}\|T^m\|\leq\sup_{i\in\mathbb{N}}\prod_{j=1+i}^{m+i}|c_j|\end{equation}

Now pick any $\epsilon>0$, find $N\in\mathbb{N}$ so that $|c_n|<\epsilon$ for all $n\geq N$, and write \begin{equation}K=\sup_{i\in\mathbb{N}}\prod_{j=1+i}^{N+i}|c_j|.\end{equation} Then \begin{equation}\|T^{N+m}\|\leq\sup_{i\in\mathbb{N}}\prod_{j=1+i}^{N+m+i}|c_j|\leq\sup_{i\in\mathbb{N}}\prod_{j=1+i}^{N+i}|c_j|\prod_{j=N+i+1}^{N+i+m}|c_j|\leq \epsilon^m K=\epsilon^{N+m}(\epsilon^{-N}K).\end{equation} Thus, $\|T^{N+m}\|^{1/(N+m)}\leq (\epsilon^{-N}K)^{1/(N+m)}\epsilon\to\epsilon$ as $m\to\infty$. Since $\epsilon>0$ was arbitrary, we have what we want.

  • $\begingroup$ Thank you very much for your assistance! $\endgroup$ – Aweygan Aug 30 '15 at 18:44

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