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In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of operators on
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
and more general spaces. Formal justification for the manipulations can be found in the framework of
holomorphic functional calculus In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''('' ...
. 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 : R(z;A)= (A-zI)^~. Among other uses, the resolvent may be used to solve the inhomogeneous
Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to sol ...
s; 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 linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
in the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of . That is, suppose there exists a simple closed curve C_\lambda in the complex plane that separates from the rest of the spectrum of . Then the
residue Residue may refer to: Chemistry and biology * An amino acid, within a peptide chain * Crop residue, materials left after agricultural processes * Pesticide residue, refers to the pesticides that may remain on or in food after they are applie ...
: -\frac \oint_ (A- z I)^~ dz defines a
projection operator In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it ...
onto the
eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of . The
Hille–Yosida theorem In functional analysis, the Hille–Yosida theorem characterizes the generators of strongly continuous one-parameter semigroups of linear operators on Banach spaces. It is sometimes stated for the special case of contraction semigroups, with the g ...
relates the resolvent through a
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the co ...
to an integral over the one-parameter
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of transformations generated by . Thus, for example, if is a
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...
, then is a one-parameter group of unitary operators. Whenever , z, >\, A\, , the resolvent of ''A'' at ''z'' can be expressed as the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the co ...
: R(z;A)= \int_0^\infty e^U(t)~dt, where the integral is taken along the ray \arg t=-\arg\lambda.


History

The first major use of the resolvent operator as a series in (cf. Liouville–Neumann series) was by Ivar Fredholm, in a landmark 1903 paper in ''Acta Mathematica'' that helped establish modern
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operator ...
. The name ''resolvent'' was given by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
.


Resolvent identity

For all in , the
resolvent set In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Definitions L ...
of an operator , we have that the first resolvent identity (also called Hilbert's identity) holds: :R(z; A) - R(w; A) = (z-w) R(z;A) R(w;A)\, . (Note that Dunford and Schwartz, cited, define the resolvent as , instead, 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 and , both defined on the same linear space, and in the following identity holds, :R(z;A) - R(z;B) = R(z;A)(A-B) R(z;B) \, .


Compact resolvent

When studying a closed
unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The te ...
: → on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, if there exists z\in\rho(A) such that R(z;A) is a
compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
, we say that has compact resolvent. The spectrum \sigma(A) of such is a discrete subset of \mathbb. If furthermore is
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...
, then \sigma(A)\subset\mathbb and there exists an orthonormal basis \_ of eigenvectors of with eigenvalues \_ respectively. Also, \ has no finite accumulation point.Taylor, p. 515.


See also

*
Resolvent set In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Definitions L ...
*
Stone's theorem on one-parameter unitary groups In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families :(U_)_ ...
*
Holomorphic functional calculus In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''('' ...
*
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
*
Compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
*
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the co ...
*
Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ...
* Liouville–Neumann series *
Decomposition of spectrum (functional analysis) The spectrum of a linear operator T that operates on a Banach space X (a fundamental concept of functional analysis) consists of all scalars \lambda such that the operator T-\lambda does not have a bounded inverse on X. The spectrum has a standar ...
* Limiting absorption principle


References

* * * . * . *{{Citation , last = Taylor , first = Michael E. , authorlink = Michael E. Taylor , title = Partial Differential Equations I , publisher = Springer-Verlag , location = New York, NY , year = 1996 , isbn = 7-5062-4252-4 Fredholm theory Formalism (deductive) Mathematical physics