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 (T(t)x - x)/t ( T ( t ) x − x ) / t as t → 0 + t \to 0^+ t → 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 ) = e x p ( t A ) T(t) = exp(tA) T ( t ) = e x p ( t A )
Resolvent of A crucial in determining C0-semigroup properties
Resolvent defined as ( λ I − A ) − 1 (λI - A)^{-1} ( λ I − A ) − 1 for λ in resolvent set of A
Plays key role in spectral theory and semigroup generation
Limit definition of generator: A x = lim t → 0 + T ( t ) x − x t Ax = \lim_{t \to 0^+} \frac{T(t)x - x}{t} A x = lim t → 0 + t T ( t ) x − x
Domain of A: [ D ( A ) ] ( h t t p s : / / w w w . f i v e a b l e K e y T e r m : d ( a ) ) = { x ∈ X : lim t → 0 + T ( t ) x − x t exists } [D(A)](https://www.fiveableKeyTerm:d(a)) = \{x \in X : \lim_{t \to 0^+} \frac{T(t)x - x}{t} \text{ exists}\} [ D ( A )] ( h ttp s : // www . f i v e ab l eKey T er m : d ( a )) = { x ∈ X : lim t → 0 + t T ( t ) x − x exists }
Closedness of A implies graph of A closed in X × X X \times X X × 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 ) = e x p ( t A ) T(t) = exp(tA) T ( t ) = e x p ( t A ) understood through power series expansion or functional calculus
Resolvent R(λ,A) connected to Laplace transform of semigroup: R ( λ , A ) = ∫ 0 ∞ e − λ t T ( t ) d t R(λ,A) = \int_0^\infty e^{-λt}T(t)dt R ( λ , A ) = ∫ 0 ∞ e − λ t T ( t ) d t 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 ∥ ≤ M / ( λ − ω ) n \|R(λ,A)^n\| \leq M/(λ-ω)^n ∥ R ( λ , A ) n ∥ ≤ M / ( λ − ω ) n for n ≥ 1 n \geq 1 n ≥ 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 ) ) ∖ { 0 } = e t σ ( A ) σ(T(t)) \setminus \{0\} = e^{tσ(A)} σ ( T ( t )) ∖ { 0 } = e t σ ( 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 R e ⟨ A x , x ∗ ⟩ ≤ 0 Re\langle Ax, x^*\rangle \leq 0 R e ⟨ A x , x ∗ ⟩ ≤ 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 (T(t)x - x)/t ( T ( t ) x − x ) / t as t → 0 + t \to 0^+ t → 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 ) = ∫ 0 ∞ e − λ t T ( t ) d t R(λ,A) = \int_0^\infty e^{-λt}T(t)dt R ( λ , A ) = ∫ 0 ∞ e − λ t T ( t ) d t for Re(λ) large enough
Utilize exponential formula to confirm A generates given semigroup
Check if T ( t ) = e x p ( t A ) T(t) = exp(tA) T ( t ) = e x p ( t A ) 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 ) = e x p ( t A ) T(t) = exp(tA) T ( t ) = e x p ( t A )
Semigroup property T ( t + s ) = T ( t ) T ( s ) T(t+s) = T(t)T(s) T ( t + s ) = T ( t ) T ( s ) equivalent to e x p ( ( t + s ) A ) = e x p ( t A ) e x p ( s A ) exp((t+s)A) = exp(tA)exp(sA) e x p (( t + s ) A ) = e x p ( t A ) e x p ( s A )
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 ) = ∫ 0 ∞ e − λ t T ( t ) d t R(λ,A) = \int_0^\infty e^{-λt}T(t)dt R ( λ , A ) = ∫ 0 ∞ e − λ t T ( t ) d t 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