Let $X$ be a complex [Banach space](/page/Banach%20Space), let $T \in \mathcal{L}(X)$, and let $I_X \in \mathcal{L}(X)$ denote the identity operator on $X$. 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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For $\lambda \in \rho(T)$, define the resolvent operator by