# Dictionary Definition

resolvent n : a liquid substance capable of
dissolving other substances; "the solvent does not change its state
in forming a solution" [syn: solvent, dissolvent, dissolver, dissolving
agent]

# Extensive Definition

In mathematics, the resolvent
formalism is a technique for applying concepts from complex
analysis to the study of the
spectrum of operators on Hilbert
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

- R(z;A)= (A-zI)^.

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.

- \operatorname R(z;A)\vert_ =

where λ corresponds to an eigenvalue of A

- A\vert \psi_\lambda \rangle = \lambda \vert \psi_\lambda \rangle

and C_\lambda is a contour in the positive
direction around the eigenvalue λ.

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
U(t)=\exp(itA) is a one-parameter group of unitary operators. The
resolvent can be expressed as the integral

- R(z;A)= \int_0^\infty e^U(t) dt.

## 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.## References

- E.I. Fredholm, "Sur une classe d'equations fonctionnelles", Acta Mathematica , 27 (1903) pp. 365–390.

resolvent in Russian: Резольвента

resolvent in German: Resolvente

# Synonyms, Antonyms and Related Words

ablative, anticoagulant, biodegradable, corrosive, cutting, decomposable, decomposing, degradable, dilapidated, diluent, dilutant, disintegrable, disintegrated, disintegrating, disintegrative, disjunctive, disruptive, dissolvent, dissolving agent,
erosive, flux, liquefacient, liquefier, moldering, ravaged, resolutive, ruinous, separative, solvent, thinner, thinning, worn