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, the resolvent may be defined as
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 can be used to directly obtain information about the spectral decomposition
of. For example, suppose is an isolated eigenvalue in the
spectrum of. That is, suppose there exists a simple closed curve
in the complex plane that separates from the rest of the spectrum of.
Then the residue
defines a projection operator onto the eigenspace of.
The Hille–Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated by. Thus, for example, if is a Hermitian, then is a one-parameter group of unitary operators. The resolvent of iA can be expressed as the Laplace transform

History

The first major use of the resolvent operator as a series in was by Ivar Fredholm, in a landmark 1903 paper in Acta Mathematica that helped establish modern operator theory.
The name resolvent was given by David Hilbert.

Resolvent identity

For all in , the resolvent set of an operator , we have that the first resolvent identity holds:
The second resolvent identity is a generalization of the first resolvent identity, above, useful for comparing the resolvents of two distinct operators. Given operators and , both defined on the same linear space, and in the following identity holds,

Compact resolvent

When studying an unbounded operator : → on a Hilbert space , if there exists such that is a compact operator, we say that has compact resolvent. The spectrum of such is a discrete subset of. If furthermore is self-adjoint, then and there exists an orthonormal basis of eigenvectors of with eigenvalues respectively. Also, has no finite accumulation point.