# Norm, Spectrum, and Adjoint of an Operator

The problem I'm trying to solve reads: Let $$\{w_k\}_{k \in \mathbb{N}} \in \ell^{\infty}$$, and define $$T: \ell^2 \to \ell^2$$, $$T\left(\{x_n\}_{n\in\mathbb{N}}\right) = \{w_nx_n\}_{n \in \mathbb{N}}$$. Compute $$||T||$$, $$\sigma(T)$$, and $$T^*$$.

So far I've come up with $$||T|| = ||\{w_k\}||_{\infty}$$ (using the definition of an operator norm) and $$\sigma(T) = \{w_n: n \in \mathbb{N}\}$$ (by computing the inverse of $$\lambda I - T$$).

I think I'm stuck on the third part, though. I started with $$\langle Tx,y \rangle = \langle x,T^*y \rangle$$ and got as far as $$\sum_{n=1}^{\infty} x_n(w_n\overline{y_n} - \overline{T^*y_n}) = 0$$. But I don't know what to do next.

Am I on the right track? And if not, where did I go wrong? I'm basically teaching myself this material because I've been too sick to go to class, so I'm not sure if I understand it or not.

• From here math.stackexchange.com/questions/1337896/… it seems that $\|T\| = \|w\|_\infty$ is correct. Commented Nov 26, 2019 at 2:17
• This question seems to address the spectrum of your given operator: math.stackexchange.com/questions/3031388/… Commented Nov 26, 2019 at 2:18
• @Math1000 Thank you! I thought $\{0\}$ seemed too simple; I computed the inverse wrong. Instead of solving $(\lambda S - ST)x = x$ for $S$, I accidentally solved $(\lambda S - T)x = x$. And that's a bit different. :) Commented Nov 26, 2019 at 3:13

Let $$x, y \in \ell^2$$. \begin{align*} \langle Tx, y \rangle &= \langle x, T^*y \rangle\\ \langle wx,y \rangle &= \langle x, T^*y \rangle\\ \sum_{n=1}^{\infty} w_nx_n\overline{y_n} &= \sum_{n=1}^{\infty} x_n\overline{T^*y_n}\\ \sum_{n=1}^{\infty} (w_nx_n\overline{y_n} - x_n\overline{T^*y_n}) &= 0\\ \sum_{n=1}^{\infty} x_n(w_n\overline{y_n} - \overline{T^*y_n}) &= 0\\ \end{align*} Note that this equation holds for all $$x,y \in \ell^2$$; in particular, it holds for $$x = \{e_i\}$$ for each $$i \in \mathbb{N}$$ and $$y$$ such that only the $$i$$th term is nonzero. It follows that for each $$i$$, we have \begin{align*} \sum_{n=1}^{\infty} (w_n\overline{y_n} - \overline{T^*y_n}) &= w_i\overline{y_i} - \overline{(T^*y)_i} =0\\ w_i \overline{y_i} &= \overline{(T^*y)_i}\\ w\overline{y} &= \overline{T^*y}\\ \overline{w}y &= T^*y \end{align*}