Is there some operator generator of an analytical semigroup with eigenvalues such that $\sum_{k} \frac{1}{|\lambda_k|} < \infty$?

Question

The following is pretty relevant for my research:

Consider the Laplacian $\Delta$ in 1-D then under certain conditions it is well known that it is diagonalizable and its eigenvalues are $\lambda_k=-k^2$, $k \in \mathbb{N}$ so that $$\sum_{k} \frac{1}{|\lambda_k|} < \infty$$ Note that the series doesn't converge in dimension $d=2,3$ as $\lambda_k =-k^{2/d}$.

Now (except for the Laplacian in 1-D) is there some other linear operator a generator of an analytical semigroup which is diagonalizable with eigenvalues $\lambda_k$ such that satisfy $$\sum_{k} \frac{1}{|\lambda_k|} < \infty ?$$