8.2 Generators of C0-semigroups

C0-semigroups are crucial in operator theory, modeling continuous-time processes. Generators of these semigroups capture their infinitesimal behavior, determining the semigroup's properties and evolution. Understanding generators is key to analyzing dynamic systems and solving abstract Cauchy problems.

This section dives into the nitty-gritty of generators, exploring their definition, properties, and relationship to C0-semigroups. We'll learn how to identify and work with generators, unraveling their role in spectral theory and semigroup generation.

Generators of C0-semigroups

Definition and Fundamental Concepts

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• Generator of a C0-semigroup determines behavior of semigroup T(t) as t approaches 0
• Generator A defined as limit of $(T(t)x - x)/t$ as $t \to 0^+$, for all x in domain of A
• Domain of A includes all x where this limit exists
• A closed, densely defined linear operator on Banach space X
• Uniquely determines C0-semigroup (Hille-Yosida theorem)
• Exponential formula relates A to T(t) through $T(t) = exp(tA)$
• Resolvent of A crucial in determining C0-semigroup properties
  • Resolvent defined as $(λI - A)^{-1}$ for λ in resolvent set of A
  • Plays key role in spectral theory and semigroup generation

Mathematical Formulation and Implications

• Limit definition of generator: $Ax = \lim_{t \to 0^+} \frac{T(t)x - x}{t}$
• Domain of A: $[D(A)](https://www.fiveableKeyTerm:d(a)) = \{x \in X : \lim_{t \to 0^+} \frac{T(t)x - x}{t} \text{ exists}\}$
• Closedness of A implies graph of A closed in $X \times X$
• Density of D(A) crucial for well-posedness of associated abstract Cauchy problems
• Hille-Yosida theorem provides necessary and sufficient conditions for A to generate a C0-semigroup
• Exponential formula $T(t) = exp(tA)$ understood through power series expansion or functional calculus
• Resolvent R(λ,A) connected to Laplace transform of semigroup: $R(λ,A) = \int_0^\infty e^{-λt}T(t)dt$ for Re(λ) large enough

Properties of Generators

Spectral and Resolvent Properties

• Generators unbounded operators with domain as proper subspace of X
• Spectrum of A contained in left half-plane of complex plane, with growth bound ω
• Resolvent set of A contains all λ with Re(λ) > ω
  • ω represents exponential growth bound of semigroup
• Resolvent operator R(λ,A) satisfies estimate $\|R(λ,A)^n\| \leq M/(λ-ω)^n$ for $n \geq 1$ and λ > ω
  • M constant related to semigroup bound
• Range of λI - A dense in X for all λ > ω
  • I represents identity operator
• Spectral mapping theorem: $σ(T(t)) \setminus \{0\} = e^{tσ(A)}$ for all t ≥ 0
  • Relates spectrum of semigroup to spectrum of generator

Analytical and Structural Characteristics

• A satisfies Hille-Yosida conditions
  • Characterize when closed operator generates C0-semigroup
• Dissipativity of A equivalent to contractivity of associated C0-semigroup
  • A dissipative if $Re\langle Ax, x^*\rangle \leq 0$ for all x in D(A) and x* in dual space
• Core theorem simplifies generator identification
  • Core: dense subspace D ⊂ D(A) invariant under T(t)
  • Sufficient to determine A on core and extend by closure
• Generator determines asymptotic behavior of semigroup
  • Stability, asymptotic stability, exponential stability related to spectral properties of A
• Perturbation theory for generators extends to semigroups
  • Bounded perturbations: A + B generates C0-semigroup if B bounded and A generates C0-semigroup
  • Relatively bounded perturbations: more general class allowing certain unbounded perturbations

Determining Generators

Computational Techniques

• Compute limit of $(T(t)x - x)/t$ as $t \to 0^+$ for x in suitable domain
• Identify maximal domain where limit exists to determine D(A)
• Verify obtained operator satisfies generator properties (closed, densely defined)
• Use core theorem to simplify process
  • Find generator on core and extend by closure
• Apply resolvent formula to compute R(λ,A) and verify properties
  • $R(λ,A) = \int_0^\infty e^{-λt}T(t)dt$ for Re(λ) large enough
• Utilize exponential formula to confirm A generates given semigroup
  • Check if $T(t) = exp(tA)$ holds for all t ≥ 0
• For specific semigroup classes, use known formulas
  • Translation group on L^p(R): A = d/dx with D(A) = W^{1,p}(R)
  • Rotation group on L^2(R^2): A = -y∂/∂x + x∂/∂y with suitable domain

Examples and Special Cases

• Heat semigroup on L^2(R^n): A = Δ (Laplacian) with D(A) = H^2(R^n)
• Wave semigroup on energy space: A = (0, I; Δ, 0) with appropriate domain
• Ornstein-Uhlenbeck semigroup: A = Δ - x⋅∇ with weighted Sobolev space domain
• Multiplication semigroup: A = multiplication by function g(x) with D(A) = {f : gf ∈ X}
• Nilpotent shift semigroup: A = d/dx with D(A) = {f ∈ W^{1,p}(0,1) : f(0) = 0}
• Analytic semigroups: A sectorial operator (spectrum in sector, resolvent bounds)
  • Example: A = -Δ with Dirichlet boundary conditions on bounded domain

C0-semigroups vs Generators

Functional Relationships

• Generator A completely determines C0-semigroup T(t) through $T(t) = exp(tA)$
• Semigroup property $T(t+s) = T(t)T(s)$ equivalent to $exp((t+s)A) = exp(tA)exp(sA)$
• Strong continuity of T(t) at t=0 related to density of D(A) in X
• Growth bound of semigroup determined by spectral bound of generator
  • ω = sup{Re(λ) : λ ∈ σ(A)}
• Resolvent of A expressible as integral involving semigroup
  • $R(λ,A) = \int_0^\infty e^{-λt}T(t)dt$ for Re(λ) > ω
• Abstract Cauchy problem u'(t) = Au(t) with u(0) = x solved by u(t) = T(t)x
  • T(t) semigroup generated by A
• Perturbation theory for semigroups studied through generator perturbations
  • Trotter-Kato theorem provides conditions for convergence of perturbed semigroups

Analytical and Practical Implications

• Generator encodes infinitesimal behavior of semigroup
  • Useful for studying local properties and short-time asymptotics
• Semigroup provides global solution operator for associated evolution equations
  • Facilitates study of long-time behavior and asymptotic properties
• Spectral mapping theorem connects spectra of A and T(t)
  • Allows inference of semigroup properties from generator spectrum
• Hille-Yosida theorem characterizes generators through resolvent bounds
  • Provides practical criteria for verifying if operator generates C0-semigroup
• Stone's theorem establishes bijection between self-adjoint operators and unitary groups
  • Special case linking generators to strongly continuous unitary groups
• Lumer-Phillips theorem characterizes generators of contractive C0-semigroups
  • Uses concept of dissipativity, important in applications to physical systems
• Trotter product formula relates semigroups generated by sum of operators to product of individual semigroups
  • Useful in approximation theory and numerical methods for evolution equations