Let $\lambda_0$ is a discrete (isolated) spectrum of a densely defined closed linear operator $\mathcal{L}: H \to H$, where $H$ is a Hilbert space. Suppose that $\lambda_0$ has a finite algebraic multiplicity $k$. Then, the adjoint operator of $\mathcal{L}^a$ has an eigenvalue $\overline{\lambda_0}$ with multiplicity $k$. Let $E_{\lambda_0}$ and $E_{\overline{\lambda_0}}^a$ be the generalized eigenspaces of $\mathcal{L}$ and $\mathcal{L}^a$ corresponding to $\lambda_0$ and $\overline{\lambda_0}$, respectively. We do not assume that $\mathcal{L}$ is self-adjoint.
Now, is it possible to have the decomposition $H=E_{\lambda_0}\oplus (E_{\overline{\lambda_0}}^a)^\perp$, where the direct sum is orthogonal?
Thank you very much!