# 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.

The **resolvent** captures the spectral properties of an operator in the analytic structure of the resolvent. Given an operator *A*, 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 *A* can be used to directly obtain information about the spectral decomposition
of *A*. For example, suppose is an isolated eigenvalue in the
spectrum of *A*. That is, suppose there exists a simple closed curve
in the complex plane
that separates from the rest of the spectrum of *A*.
Then the residue

defines a projection operator onto the eigenspace of *A*.

The Hille-Yosida theorem relates the resolvent to an integral over the one-parameter group of transformations generated by *A*. Thus, for example, if *A* is Hermitian, then is a one-parameter group of unitary operators. The resolvent can be expressed as the integral

## History

The first major use of the resolvent operator 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** (also called Hilbert's identity) holds:^{[1]}

(Note that Dunford and Schwartz define the resolvent as so that the formula above is slightly different from theirs.)

The **second resolvent identity** is a generalization of the first resolvent identity, useful for comparing the resolvents of two distinct operators. Given operators and , both defined on the same linear space, and in it holds that:^{[2]}

## 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.^{[3]}

## See also

- Resolvent set
- Stone's theorem
- Holomorphic functional calculus
- Spectral theory
- Compact operator
- Unbounded operator

## References

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