In my functional analysis class I was asked to find the spectrum of the following multiplication operator $$A: L^2[0,2\pi]\rightarrow L^2[0,2\pi]\ \ \ \ (Ax)(t)=\sin(3/t)x(t)$$
Here are my thoughts: I understand that this operator is self-adjoint and bounded: $\|A\|=1$, hence $\sigma(A)\subset [-1,1]$. From my lecture notes I also grabbed the fact that if $A$ is self-adjoint, then $\|A\|=\max \{ |\lambda|, \lambda\in\sigma(A) \} $, in my case it means that either $-1$ or $1$ lies in spectrum. But I don't know what should I do next.
I feel like may be I should check it for compactness and use Hilbert-Schmidt theorem, but I didn't figure it out. So I'm asking for a hints what should I do next, thanks!
Edit: Apparently, it's well-known, that the spectrum of such operator coincide with the family of its "essential values" $$\textrm{Ess}\ \varphi = \{\lambda: \forall \varepsilon>0\ \mu(t: |\varphi(t)-\lambda|<\varepsilon)>0 \} $$ And as soon as $\sin(3/t)$ is continuous outside of zero neighborhood, it's essential values are $[-1,1]$ so the spectrum $\sigma(A)$ is also $[-1,1]$. If someone is familiar with the consept, I'll be gratefull for verification!