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 functional. 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 $λ$ 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 $λ$ from the rest of the spectrum of $A$ . Then the residue

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

defines a projection operator onto the $λ$ eigenspace of $A$ .

Further information: Frobenius covariant and Holomorphic functional calculus

The Hille-Yosida theorem relates the resolvent through a Laplace transform 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 their Laplace transform integral

R(z;A)=\int _{0}^{\infty }e^{{-zt}}U(t)dt~.

History

The first major use of the resolvent operator as a series in $A$ (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 $z, w$ in $ρ (A)$ , 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)=(z-w)R(z;A)R(w;A)\,.

(Note that Dunford and Schwartz, cited, define the resolvent as $(zI -A) -1$ , 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 $A$ and $B$ , both defined on the same linear space, and $z$ in $ρ (A)\cap ρ (B)$ the following identity holds,^[2]

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

Compact resolvent

When studying an unbounded operator $A$ : $H$ → $H$ 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]

References

↑ Dunford and Schwartz, Vol I, Lemma 6, p. 568.
↑ Hille and Phillips, Theorem 4.82, p. 126
↑ Taylor, p. 515.

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

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Resolvent formalism

History

Resolvent identity

Compact resolvent

See also

References