spectrum of compact forward weighted shift

Question

Let $H$ be an infinite dimensional, separable complex Hilbert space with orthonormal basis $(e_n)_n$.

Let $T$ be the unilateral forward weighted shift so that $Te_n=t_n e_{n+1}$.

Suppose $T$ is compact. Find $\sigma(T)$.

Things I know:

I know that $\sigma(T)$ is always compact and hence closed. So I'm suspecting that $\sigma(T)$ is the closure of some set, maybe the eigenvalues?

Other than that, I'm not sure how to continue.

Any help or ideas will be appreciated!

Answer 1

Hints: as mentioned in the comments, $t_n\to 0$. The weights I assume are positive by definition. Now, since $\|Tx\|^²=\sum|t_nx_n|^2$ it is not hard to show that $\|T\|=\max \{t_n\}.$ In fact, inductively, show that

$\tag1 \|T^n\|=\max_k\{t_kt_{k+1}\cdots t_{k+n-1}\}$

Now, choose $n$ large enough so that $t_n<\epsilon$ whenever $n\ge N$ so

$\tag2 t_{k+N}\cdots t_{k+n-1}\le \epsilon^{n-N-1}$

Finally, combine $(1)$ and $(2)$ with the spectral radius formula to show that $\sigma (T)=0.$