Resolvent formalism

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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

References

  1. Dunford and Schwartz, Vol I, Lemma 6, p568.
  2. Hille and Phillips, Theorem 4.82, p. 126
  3. Taylor, p515.
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