Let $H$ be a complex Hilbert space and $N: D(N) \subset H \to H$ be normal densely defined operator such that \begin{equation}\tag{1} (s+N)^{-1}\in \mathcal{B}(H)\,\,\text{and}\,\,(s+N)^{-1}(D(N))\subset D(N)\,\,\, \forall s >0. \end{equation} Let $K$ be a compact subset in $\mathbb{C}$ and denote
$$N_K := \int_{\sigma(N)} \lambda \chi_K(\lambda)dE(\lambda)= \int_{K}\lambda dE(\lambda).$$ In view of the spectral theorem, we know that such $E$ exists and it is called the resolution of the identity.
I wonder whether the operator $N_K$ satisfies (1). The first part is easy to verify.
Thank you for any hint.