Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{D} \to \mathscr{H}$ be positive and negative (unbounded) operators, respectively, meaning that $$ \langle Pv,v \rangle \geq 0 \quad \text{and} \quad \langle Nv,v \rangle \leq 0 $$ for all $v \in \mathscr{D}$. Moreover, assume that $P$ and $N$ commute.

Consider the operator sum $X := P + N$. If $\mathscr{D}$ decomposes as a finite sum of eigenspaces for $X$, is it true that $\mathscr{D}$ also decomposes into eigenspaces for $P$ and $N$, respectively?

Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{D} \to \mathscr{H}$ be positive and negative (unbounded) operators, respectively, meaning that $$ \langle Pv,v \rangle \geq 0 \quad \text{and} \quad \langle Nv,v \rangle \leq 0 $$ for all $v \in \mathscr{D}$. Moreover, assume that $P$ and $N$ commute, and $P \neq -N$.

Consider the operator sum $X := P + N$. If $\mathscr{D}$ decomposes as a finite sum of eigenspaces for $X$, is it true that $\mathscr{D}$ also decomposes into eigenspaces for $P$ and $N$, respectively?