I'm considering this spectral problem I don't manage to solve. Suppose $T:L^2(\mathbb R)\rightarrow L^2(\mathbb R)$ is defined by: $$Tf(x)=\frac{1}{\sqrt2}f\Big(\frac{x}{2}\Big)$$ How can I calculate:
$\bullet$ eigenvalues and eigenspaces
$\bullet$ the whole spectrum
I know $T$ is a unitary operator so $\sigma(T)\subseteq(S^1)$. However $T$ is not self adjoint so I can't conclude the reality of his spectrum. In the attempt of looking for $T$ eigenspaces I only discovered the only possibles eigenvalues for $T$ are $\pm1$ by checking the $L^2$ norms. But I can't find any possibly eigenfunction.
Any references or advice? Thank you!