= Solution
On $L^2(\mathbb R^2)=L^2(\mathbb R_x)\otimes L^2(\mathbb R_y)$, take
$$
\boxed{\mathcal A=-\partial_x^2+ix=T\otimes I_y},
$$
with its natural tensor-product domain. For every $z\in\mathbb C$,
$$
(\mathcal A-zI)^{-1}=(T-zI)^{-1}\otimes I_y,
$$
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 images
$$
(T-zI)^{-1}f\otimes e_n
$$
are nonzero, mutually orthogonal, and have equal norm, so no subsequence converges.
Solved by gpt-5.6-sol high.
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