2.3 Resolvent and resolvent set

The resolvent and resolvent set are key concepts in understanding the spectrum of linear operators. They provide a powerful tool for analyzing an operator's behavior and spectral properties. By studying the resolvent, we gain insights into the operator's invertibility and its relationship to complex numbers.

The resolvent set complements the spectrum, forming an open subset of the complex plane. This relationship allows us to classify different types of spectra and understand how the resolvent's norm behaves near spectral points. These ideas are fundamental for applying operator theory in various fields of mathematics and physics.

Resolvent and Resolvent Set

Definitions and Properties

Top images from around the web for Definitions and Properties

• 

  Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane ... View original

  Is this image relevant?

• 

  Some Result of Stability and Spectra Properties on Semigroup of Linear Operator View original

  Is this image relevant?

• 

  Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane ... View original

  Is this image relevant?

• 

  Some Result of Stability and Spectra Properties on Semigroup of Linear Operator View original

  Is this image relevant?

1 of 2

Top images from around the web for Definitions and Properties

• 

  Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane ... View original

  Is this image relevant?

• 

  Some Result of Stability and Spectra Properties on Semigroup of Linear Operator View original

  Is this image relevant?

• 

  Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane ... View original

  Is this image relevant?

• 

  Some Result of Stability and Spectra Properties on Semigroup of Linear Operator View original

  Is this image relevant?

1 of 2

• Resolvent of linear operator T on Banach space X defined as $R(\lambda,T) = (\lambda I - T)^{-1}$
  • λ denotes complex number
  • I represents identity operator
• Resolvent set ρ(T) encompasses all complex numbers λ where R(λ,T) exists as bounded linear operator on X
• ρ(T) forms open subset of complex plane
• Complement of ρ(T) constitutes spectrum of operator T
• R(λ,T) functions as analytic function of λ on ρ(T)
• Norm of resolvent $\|R(\lambda,T)\|$ approaches infinity as λ nears boundary of resolvent set
• Resolvent set provides insights into operator's behavior and spectral properties

Mathematical Characteristics

• Resolvent R(λ,T) exhibits analyticity in λ over ρ(T)
  • Allows for power series expansions and contour integration techniques
• Norm of resolvent $\|R(\lambda,T)\|$ inversely related to distance from λ to spectrum
  • Provides measure of how close λ lies to spectral values
• Resolvent satisfies resolvent identity
  • $R(\lambda,T) - R(\mu,T) = (\mu-\lambda)R(\lambda,T)R(\mu,T)$ for λ, μ in ρ(T)
• Resolvent set ρ(T) characterized by boundedness and existence of (λI - T)^(-1)
  • Ensures well-defined inverse operator

Calculating the Resolvent

Computational Process

• Form operator (λI - T) for given complex number λ
• Determine invertibility of (λI - T)
  • Check injectivity and surjectivity on domain of T
• Compute inverse (λI - T)^(-1) if invertible
  • Resulting inverse constitutes resolvent R(λ,T)
• Finite-dimensional operators utilize matrix inversion techniques
  • (Gaussian elimination, LU decomposition)
• Infinite-dimensional operators may require advanced functional analysis methods
  • (Spectral theory, operator decomposition)
• Non-existence of resolvent for certain λ values indicates spectrum of T

Practical Examples

• Calculate resolvent for 2x2 matrix operator
  • T = [[1, 2], [3, 4]], λ = 5
  • (5I - T) = [[4, -2], [-3, 1]]
  • R(5,T) = (5I - T)^(-1) = [[1/10, 1/5], [3/10, -2/5]]
• Determine resolvent for differential operator
  • T = d/dx on C[0,1], λ ≠ 0
  • R(λ,T)f(x) = (1/λ)∫[0,x] e^(λ(x-t))f(t)dt
• Compute resolvent for multiplication operator
  • (Mφf)(x) = φ(x)f(x) on L^2(Ω)
  • R(λ,Mφ)f = f/(λ-φ) when λ ∉ range(φ)

Resolvent vs Spectrum

Complementary Relationship

• Spectrum σ(T) defined as complement of resolvent set in complex plane
  • σ(T) = ℂ \ ρ(T)
• R(λ,T) exists as bounded linear operator if and only if λ ∉ σ(T)
• Spectrum classification based on resolvent behavior
  • Point spectrum (eigenvalues)
  • Continuous spectrum
  • Residual spectrum
• Spectral radius equals limit of reciprocal of resolvent norm as λ approaches infinity
  • $r(T) = \lim_{|\lambda| \to \infty} \frac{1}{\|R(\lambda,T)\|}$
• Resolvent singularities correspond to spectral points
  • Isolated singularities often indicate eigenvalues
• Resolvent identity relates resolvents at different complex points
  • $R(\lambda,T) - R(\mu,T) = (\mu-\lambda)R(\lambda,T)R(\mu,T)$

Spectral Analysis Examples

• Analyze spectrum of shift operator on l^2
  • T((x_1, x_2, x_3, ...)) = (0, x_1, x_2, ...)
  • σ(T) = {λ : |λ| ≤ 1}
  • ρ(T) = {λ : |λ| > 1}
• Examine spectrum of multiplication operator
  • (Mφf)(x) = φ(x)f(x) on L^2(Ω)
  • σ(Mφ) = closure of range(φ)
• Study spectrum of compact operator
  • σ(T) \ {0} consists only of eigenvalues
  • 0 may or may not be an eigenvalue

Applications of the Resolvent

Operator Analysis Tools

• Functional calculus for bounded linear operators defined using resolvent
  • f(T) = (1/2πi)∫[γ] f(λ)R(λ,T)dλ for suitable contour γ
• Spectral mapping theorem relates spectrum of f(T) to image of σ(T) under analytic function f
  • σ(f(T)) = f(σ(T)) for f analytic on domain containing σ(T)
• Operator exponential and semigroup theory utilize resolvent
  • e^(tT) = (1/2πi)∫[γ] e^(tλ)R(λ,T)dλ
• Resolvent norm behavior near spectrum informs growth of $\|T^n\|$ as n approaches infinity
  • Relates to spectral radius and operator norm
• Compactness of operator linked to compactness of its resolvent
  • T compact ⇔ R(λ,T) compact for some (all) λ in ρ(T)
• Perturbation theory studies resolvent changes under small operator perturbations
  • Analyzes stability of spectral properties

Practical Applications

• Quantum mechanics uses resolvent to study energy spectra of Hamiltonians
  • Green's functions in scattering theory
• Control theory employs resolvent for stability analysis of linear systems
  • Nyquist stability criterion
• Partial differential equations utilize resolvent for solving inhomogeneous problems
  • Heat equation: u_t = Δu + f, solution involves resolvent of Laplacian
• Numerical analysis uses resolvent for iterative methods in linear algebra
  • Krylov subspace methods (GMRES, Arnoldi iteration)