and let $I_H\in\mathcal L(H)$ denote the identity operator on $H$. Let $\varphi\in L^\infty(\Omega,\mathcal F,\mu)$. Let $M_\varphi\in\mathcal L(H)$ be the bounded multiplication operator
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\begin{align*}
M_\varphi f=\varphi f,\qquad f\in H.
\end{align*}
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For $T\in\mathcal L(H)$, let
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\begin{align*}
\rho(T):=\{\lambda\in\mathbb C:T-\lambda I_H\text{ is boundedly two-sided invertible in }\mathcal L(H)\}
\end{align*}
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denote the resolvent set, let $\sigma(T):=\mathbb C\setminus\rho(T)$ denote the spectrum, and let
denote the point spectrum. Define the essential range of $\varphi$ by
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\begin{align*}
\operatorname{ess\,ran}(\varphi):=\{z\in\mathbb C:\mu(\{\omega\in\Omega:|\varphi(\omega)-z|<\varepsilon\})>0\text{ for every }\varepsilon>0\}.
\end{align*}