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

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

Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville–Neumann series.

The resolvent of A can be used to directly obtain information about the spectral decomposition of A. For example, suppose λ is an isolated eigenvalue in the spectrum of A. That is, suppose there exists a simple closed curve ${\displaystyle C_{\lambda }}$ in the complex plane that separates λ from the rest of the spectrum of A. Then the residue

${\displaystyle -{\frac {1}{2\pi i}}\oint _{C_{\lambda }}(A-zI)^{-1}~dz}$

defines a projection operator onto the λ eigenspace of A. The Hille–Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated by A.^[1] Thus, for example, if A is a skew-Hermitian matrix, then U(t) = exp(tA) is a one-parameter group of unitary operators. Whenever ${\displaystyle |z|>\|A\|}$, the resolvent of A at z can be expressed as the Laplace transform

${\displaystyle R(z;A)=\int _{0}^{\infty }e^{-zt}U(t)~dt,}$

where the integral is taken along the ray ${\displaystyle \arg t=-\arg \lambda }$.^[2]