Let $X$ be a nonzero complex [Banach space](/page/Banach%20Space), let $I_X \in \mathcal{L}(X)$ be the identity operator, and let $T \in \mathcal{L}(X)$. Define the resolvent set by
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\begin{align*}
\rho(T)=\{\lambda\in\mathbb C:T-\lambda I_X \text{ is invertible in } \mathcal{L}(X)\}.
\end{align*}
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Define the resolvent map $R_T:\rho(T)\to\mathcal{L}(X)$ by