Resolvent formalism

Template:Short description 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. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus.

The resolvent captures the spectral properties of an operator in the analytic structure of the functional. Given an operator Template:Mvar, the resolvent may be defined as

$R(z;A)=(A-zI{)}^{-1}.$

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 Template:Mvar can be used to directly obtain information about the spectral decomposition of Template:Mvar. For example, suppose Template:Mvar is an isolated eigenvalue in the spectrum of Template:Mvar. That is, suppose there exists a simple closed curve $C_{\lambda }$ in the complex plane that separates Template:Mvar from the rest of the spectrum of Template:Mvar. Then the residue

$-\frac{1}{2\pi i}{\oint }_{C_{\lambda }}(A-zI{)}^{-1}dz$

defines a projection operator onto the Template:Mvar eigenspace of Template:Mvar. The Hille–Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated by Template:Mvar. Thus, for example, if Template:Mvar is a skew-Hermitian matrix, then Template:Math is a one-parameter group of unitary operators. Whenever $|z|>\Vert A\Vert$, the resolvent of A at z can be expressed as the Laplace transform

$R(z;A)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-zt}U(t)dt,$

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 Template:Mvar (cf. Liouville–Neumann series) 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 Template:Math in Template:Math, the resolvent set of an operator Template:Mvar, 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 Template:Math, 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 Template:Mvar and Template:Mvar, both defined on the same linear space, and Template:Mvar in Template:Math the following identity holds,

$R(z;A)-R(z;B)=R(z;A)(B-A)R(z;B).$

A one-line proof goes as follows:

$(A-zI{)}^{-1}-(B-zI{)}^{-1}=(A-zI{)}^{-1}((B-zI)-(A-zI))(B-zI{)}^{-1}=(A-zI{)}^{-1}(B-A)(B-zI{)}^{-1}.$

Compact resolvent

When studying a closed unbounded operator Template:Mvar: Template:Mvar → Template:Mvar on a Hilbert space Template:Mvar, if there exists $z\in \rho (A)$ such that $R(z;A)$ is a compact operator, we say that Template:Mvar has compact resolvent. The spectrum $\sigma (A)$ of such Template:Mvar is a discrete subset of $\mathbb{ℂ}$. If furthermore Template:Mvar is self-adjoint, then $\sigma (A)\subset \mathbb{ℝ}$ and there exists an orthonormal basis $\{v_i{\}}_{i\in \mathbb{\mathbb{N}}}$ of eigenvectors of Template:Mvar with eigenvalues $\{{\lambda }_i{\}}_{i\in \mathbb{\mathbb{N}}}$ respectively. Also, $\{{\lambda }_i\}$ has no finite accumulation point.

See also

• Resolvent set
• Stone's theorem on one-parameter unitary groups
• Holomorphic functional calculus
• Spectral theory
• Compact operator
• Laplace transform
• Fredholm theory
• Liouville–Neumann series
• Decomposition of spectrum (functional analysis)
• Limiting absorption principle

References

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