Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space) with [inner product](/page/Inner%20Product) $(\cdot,\cdot)_H$ linear in the first argument, and let $\|x\|_H:=(x,x)_H^{1/2}$ denote its norm. Let $\mathcal L(H)$ denote the space of bounded linear operators from $H$ to $H$, equipped with the operator norm $\|\cdot\|_{\mathcal L(H)}$. Write $I:=I_H:H\to H$ for the identity operator on $H$. For $T\in\mathcal L(H)$, write $T^*$ for the Hilbert-space adjoint of $T$, characterized by $(Tx,y)_H=(x,T^*y)_H$ for all $x,y\in H$. For $S\in\mathcal L(H)$, define its kernel and range by

\begin{align*} \ker(S):=\{x\in H:Sx=0\} \end{align*}

and

\begin{align*} \operatorname{Range}(S):=S(H)=\{Sx:x\in H\}. \end{align*}

For a linear subspace $M\subset H$, write $M^\perp:=\{y\in H:(x,y)_H=0\text{ for every }x\in M\}$ for its orthogonal complement, and write $\overline M$ for its closure in the norm topology of $H$. Let $A\in\mathcal L(H)$ be self-adjoint, meaning $A^*=A$. Define the resolvent set of $A$ by

\begin{align*} \rho(A):=\{\lambda\in\mathbb C:A-\lambda I\in\mathcal L(H)\text{ is bijective and }(A-\lambda I)^{-1}\in\mathcal L(H)\}. \end{align*}

Define the spectrum of $A$ by

\begin{align*} \sigma(A):=\mathbb C\setminus\rho(A). \end{align*}

Then

\begin{align*} \sigma(A)\subset\mathbb R. \end{align*}

More precisely, if $\lambda\in\mathbb C\setminus\mathbb R$, then $A-\lambda I$ is bijective, its inverse $(A-\lambda I)^{-1}:H\to H$ is bounded, and

\begin{align*} \|(A-\lambda I)^{-1}\|_{\mathcal L(H)}\leq \frac{1}{|\operatorname{Im}\lambda|}. \end{align*}