Let $H$ be a nonzero complex [Hilbert space](/page/Hilbert%20Space), let $T \in \mathcal{L}(H)$ be normal, and let $E:\mathcal{B}(\sigma(T))\to\mathcal{L}(H)$ be the spectral measure of $T$. Then

\begin{align*} \sigma(T)=\operatorname{supp}E. \end{align*}

Here $\operatorname{supp}E=\{\lambda\in\sigma(T): E(\sigma(T)\cap B(\lambda,\varepsilon))\ne 0 \text{ for every } \varepsilon>0\}$. For a bounded operator $A$ on a complex [Banach space](/page/Banach%20Space), $\rho(A)$ denotes its resolvent set, so $\rho(A)=\mathbb C\setminus\sigma(A)$. For a complex normed space $Y$, $I_Y:Y\to Y$ denotes the identity operator, and when the space is clear from context we write $I$.

More generally, let $(X,\mathcal{A},\mu)$ be a semi-finite [measure space](/page/Measure%20Space), let $\varphi:X\to\mathbb{C}$ be a bounded $\mathcal{A}$-measurable function, and let $M_\varphi\in\mathcal{L}(L^2(X,\mu))$ be the multiplication operator $f\mapsto \varphi f$ on $L^2(X,\mu)$. Then

\begin{align*} \sigma(M_\varphi)=\operatorname{ess\,ran}(\varphi), \end{align*}

where the closed essential range is $\operatorname{ess\,ran}(\varphi)=\{\lambda\in\mathbb C:\mu(\{x\in X:|\varphi(x)-\lambda|<\varepsilon\})>0 \text{ for every } \varepsilon>0\}$.