Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space) with [inner product](/page/Inner%20Product) $(\cdot,\cdot)_H$, linear in the first variable and conjugate-linear in the second variable. Let $T \in \mathcal{L}(H)$ be self-adjoint with respect to $(\cdot,\cdot)_H$, and let $\lambda \in \mathbb{C}$. Define the eigenspace

\begin{align*} E_\lambda := \ker(T-\lambda I), \end{align*}

where $I \in \mathcal{L}(H)$ is the identity operator on $H$. Then $E_\lambda^\perp$ is invariant under $T$; equivalently,

\begin{align*} T(E_\lambda^\perp) \subset E_\lambda^\perp. \end{align*}