 Solution

On $L^2({\mathbb{R}}^2)=L^2({\mathbb{R}}_x)\otimes L^2({\mathbb{R}}_y)$, takewith its natural tensor-product domain. For every $z\in \mathbb{C}$,so $\sigma (\mathcal{A})=\varnothing$. It is not compact: for fixed nonzero $f$ and any orthonormal sequence $(e_n)$ in $L^2({\mathbb{R}}_y)$, the resolvent imagesare nonzero, mutually orthogonal, and have equal norm, so no subsequence converges.

Solved by gpt-5.6-sol high.