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

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

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

$\frac{-1}{2\pi i} \oint_{C_\lambda} (A- z I)^{-1}~ dz$

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

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

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.

Resolvent identity

For all $z, w$ in $\rho(A)$, the resolvent set of an operator $A$, we have that the first resolvent identity (also called Hilbert's identity) holds:[1]

$R(z; A) - R(w; A) = (w-z) R(z;A) R(w;A)\, .$

(Note that Dunford and Schwartz define the resolvent as $(zI-A)^{-1}$ 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 $A$ and $B$, both defined on the same linear space, and $z$ in $\rho(A) \cap \rho(B)$ it holds that:[2]

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

Compact resolvent

When studying an unbounded operator $A:H\to H$ on a Hilbert space $H$, if there exists $z\in\rho(A)$ such that $R(z;A)$ is a compact operator, we say that $A$ has compact resolvent. The spectrum $\sigma(A)$ of such $A$ is a discrete subset of $\mathbb{C}$. If furthermore $A$ is self-adjoint, then $\sigma(A)\subset\mathbb{R}$ and there exists an orthonormal basis $\{v_i\}_{i\in\mathbb{N}}$ of eigenvectors of $A$ with eigenvalues $\{\lambda_i\}_{i\in\mathbb{N}}$ respectively. Also, $\{\lambda_i\}$ has no finite accumulation point.[3]