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
- ↑ 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: Spinger-Verlag, ISBN 0-387-07558-5.
- Taylor, Michael E. (1996), Partial Differential Equations I, New York, NY: Spinger-Verlag, ISBN 7-5062-4252-4