: Mathematics: resolvent formalism

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resolvent formalism

In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus.
The resolvent captures the spectral properties of an operator in the analytic structure of the functional. Given an operator $A$ , the resolvent may be defined as

${\\displaystyle R(z;A)=(A-zI)^{-1}~.}$

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