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
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 in the complex plane that separates λ from the rest of the spectrum of A. Then the residue
The Hille-Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated by A. Thus, for example, if A is Hermitian, then U(t) = exp(itA) is a one-parameter group of unitary operators. The resolvent can be expressed as their Laplace transform integral
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.
(Note that Dunford and Schwartz, cited, define the resolvent as , so that the formula above differs in sign from theirs.)
The second resolvent identity is a generalization of the first resolvent identity, above, useful for comparing the resolvents of two distinct operators. Given operators A and B, both defined on the same linear space, and z in the following it holds,
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.
- Resolvent set
- Stone's theorem on one-parameter unitary groups
- Holomorphic functional calculus
- Spectral theory
- Compact operator
- Unbounded operator
- Laplace transform
- Dunford and Schwartz, Vol I, Lemma 6, p568.
- Hille and Phillips, Theorem 4.82, p. 126
- Taylor, p515.
- Dunford, Nelson; Schwartz, Jacob T. (1988), Linear Operators, Part I General Theory, Hoboken, NJ: Wiley-Interscience, ISBN 0471608483
- Fredholm, Erik I. (1903), "Sur une classe d'equations fonctionnelles", Acta Mathematica 27: 365–390, doi:10.1007/bf02421317
- Hille, Einar; Phillips, Ralph S. (1957), Functional Analysis and Semi-groups, Providence: American Mathematical Society, ISBN 9780821810316.
- Kato, Tosio (1980), Perturbation Theory for Linear Operators (2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-07558-5.
- Taylor, Michael E. (1996), Partial Differential Equations I, New York, NY: Springer-Verlag, ISBN 7-5062-4252-4