The resolvent operator is a crucial concept in functional analysis that provides insight into the spectral properties of bounded linear operators. It is defined as the operator $(A - heta I)^{-1}$, where $A$ is a bounded linear operator, $ heta$ is a complex number not in the spectrum of $A$, and $I$ is the identity operator. Understanding the resolvent allows one to analyze how operators behave, especially in relation to their spectrum and eigenvalues, and it plays a significant role in applications involving differential and integral operators.
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