Glossary

8.2 Generators of C0-semigroups

5 min read•august 16, 2024

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

Mathematical Formulation and Implications

  • Limit definition of generator: Ax=lim⁡t→0+T(t)x−xtAx = \lim_{t \to 0^+} \frac{T(t)x - x}{t}
  • Domain of A: [D(A)](https://www.fiveableKeyTerm:d(a))={x∈X:lim⁡t→0+T(t)x−xt exists}[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×XX \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)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)=∫0∞e−λtT(t)dtR(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
  • of A contained in left half-plane of complex plane, with 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 for n≥1n \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
  • : σ(T(t))∖{0}=etσ(A)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 generates C0-semigroup
  • of A equivalent to contractivity of associated C0-semigroup
    • A dissipative if Re⟨Ax,x∗⟩≤0Re\langle Ax, x^*\rangle \leq 0 for all x in D(A) and x* in dual space
  • 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
    • , , related to spectral properties of A
  • 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 as t→0+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)=∫0∞e−λtT(t)dtR(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)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)T(t) = exp(tA)
  • Semigroup property T(t+s)=T(t)T(s)T(t+s) = T(t)T(s) equivalent to exp((t+s)A)=exp(tA)exp(sA)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)=∫0∞e−λtT(t)dtR(λ,A) = \int_0^\infty e^{-λt}T(t)dt for Re(λ) > ω
  • Abstract 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 unitary groups
  • 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

Key Terms to Review (28)

A: In the context of operator theory, 'a' typically represents an element related to the generators of C0-semigroups and the Hille-Yosida theorem. It often denotes a densely defined linear operator on a Banach space, which plays a crucial role in the characterization and properties of C0-semigroups of linear operators. Understanding 'a' is key to grasping how these operators evolve over time and their connection to initial value problems.
Abstract Initial Value Problem: An abstract initial value problem involves finding a solution to a differential equation within a specific framework, usually described in terms of a linear operator on a Banach space. This concept is crucial in understanding how C0-semigroups are generated, as it connects the existence and uniqueness of solutions to the properties of the underlying operators. Solving these problems often requires analyzing the generator of a C0-semigroup, leading to a broader understanding of the dynamics of linear systems.
Asymptotic Stability: Asymptotic stability refers to the property of a dynamical system where, if the system starts close to a stable equilibrium point, it will eventually converge to that point as time progresses. This concept is crucial in understanding how systems behave over time, especially in the context of solutions to differential equations and their long-term behavior.
Banach space: A Banach space is a complete normed vector space, meaning that it is a vector space equipped with a norm that allows for the measurement of vector lengths and distances, and every Cauchy sequence in the space converges to a limit within that space. This concept is fundamental in functional analysis as it provides a framework for studying various operators and their properties in a structured way.
Cauchy Problem: The Cauchy problem refers to a specific type of initial value problem for partial differential equations, where the solution is sought for an equation defined on a domain with given initial conditions. It’s crucial in understanding the behavior of solutions to these equations, especially in the context of evolution equations and semigroups. The Cauchy problem plays a key role in analyzing the generators of $C_0$-semigroups, helping to establish existence, uniqueness, and continuous dependence of solutions over time.
Closed Operator: A closed operator is a linear operator defined on a subset of a Hilbert space that has the property that if a sequence of points converges in the space and the corresponding images under the operator converge, then the limit point is also in the operator's range. This concept is essential for understanding how operators behave in various contexts, including their domains and relationships with unbounded linear operators.
Core Theorem: The core theorem provides a fundamental result in the theory of C0-semigroups, connecting the generator of a semigroup to its properties and behavior. It establishes that for a strongly continuous semigroup, the generator is densely defined and closed, allowing one to analyze the long-term behavior of the semigroup by understanding its generator. This connection is vital for solving differential equations and studying evolution equations in functional analysis.
D(a): The term d(a) refers to the domain of an unbounded operator 'a' in functional analysis, specifically when considering its adjoint and properties related to generators of C0-semigroups. It is crucial to understand that the domain indicates the set of elements for which the operator is defined, highlighting its potential limitations and the importance of adjoint operators in extending these definitions. The relationship between d(a) and these concepts plays a significant role in determining the continuity and boundedness of operators and their actions on various function spaces.
Densely defined: A densely defined operator is one that is defined on a dense subset of a Banach space, meaning that the operator's domain intersects every non-empty open subset of the space. This property is crucial in understanding the behavior of operators, particularly in relation to their closure and the existence of extensions to larger spaces.
Differential Operators: Differential operators are mathematical constructs that involve the differentiation of functions, commonly expressed as a polynomial of differential operators applied to a function. They play a crucial role in various mathematical areas, including analysis and partial differential equations, by transforming functions and their derivatives into new functions. Understanding how these operators work is essential for studying properties of solutions to differential equations, as well as for analyzing the behavior of semigroups generated by these operators.
Dissipativity: Dissipativity refers to a property of a system or operator where it absorbs energy over time, leading to a decrease in the total energy of the system. In the context of generators of C0-semigroups, dissipativity indicates that the generator's action causes energy dissipation, which is crucial for the stability and long-term behavior of the associated semigroup.
Exponential Formula: The exponential formula is a mathematical expression that relates strongly continuous semigroups and their generators, particularly in the context of the evolution of dynamical systems. It provides a way to express the solution to linear evolution equations in terms of semigroups, where the exponential function plays a crucial role in connecting the time parameter with the generator of the semigroup. This formula highlights how the action of a semigroup can be described using exponentials, often taking the form $e^{tA}$, where $A$ is the generator of the semigroup and $t$ is time.
Exponential stability: Exponential stability refers to a property of a dynamical system where the solutions converge to an equilibrium point at an exponential rate. This means that not only does the system return to its equilibrium, but it does so quickly and in a way that can be quantitatively described, often involving the decay of perturbations over time. In the context of generators of C0-semigroups, it is crucial for understanding how the behavior of solutions relates to the underlying linear operator governing the system's evolution.
Generator of a c0-semigroup: The generator of a c0-semigroup is a linear operator that describes the behavior of the semigroup as time approaches zero. It is typically denoted by $A$ and is defined in terms of the limit of the difference quotient of the semigroup. Understanding the generator is crucial because it helps to analyze the properties of the semigroup, including its continuity, differentiability, and how it evolves over time.
Growth bound: A growth bound refers to a limit on how much the norm of the semigroup can grow over time, typically expressed as an exponential bound in the form $$||T(t)|| \leq Me^{\omega t}$$ for some constants M and \( \omega \). Understanding growth bounds is essential as it ensures that solutions to the associated evolution equations do not blow up and provides insights into the long-term behavior of the system being studied.
Hilbert Space: A Hilbert space is a complete inner product space that provides a framework for discussing geometric concepts in infinite-dimensional spaces. It extends the notion of Euclidean spaces, allowing for the study of linear operators, bounded linear operators, and their properties in a more general context.
Hille-Yosida Theorem: The Hille-Yosida Theorem is a fundamental result in the theory of semigroups of linear operators, which provides necessary and sufficient conditions for a strongly continuous semigroup to be associated with a densely defined linear operator. This theorem connects the existence of strongly continuous semigroups with the properties of their generators, paving the way for applications in various areas such as partial differential equations and functional analysis.
Lumer-Phillips Theorem: The Lumer-Phillips Theorem is a fundamental result in functional analysis that characterizes the generators of strongly continuous semigroups, known as C0-semigroups. It establishes necessary and sufficient conditions for an operator to be the generator of a C0-semigroup, linking the concepts of closed densely defined operators and their semigroups. This theorem plays a crucial role in the study of linear evolution equations and the stability of dynamical systems.
Multiplication Operators: Multiplication operators are linear operators defined on a function space, where each function is multiplied by a fixed function, resulting in another function within that space. This concept is crucial when discussing generators of C0-semigroups, as multiplication operators can serve as examples of densely defined operators that generate such semigroups. Understanding how these operators interact with the structure of function spaces helps to grasp their role in the broader context of functional analysis and operator theory.
Perturbation Theory: Perturbation theory is a mathematical approach used to analyze how a small change in a system's parameters affects its properties, particularly eigenvalues and eigenvectors. It plays a crucial role in understanding stability and the behavior of operators under slight modifications, making it essential for various applications in spectral theory and operator analysis.
Resolvent: The resolvent of an operator is a powerful tool that helps us understand the operator's behavior by relating it to complex numbers. Specifically, for a bounded linear operator $T$, the resolvent is defined as the operator $(T - au I)^{-1}$, where $ au$ is a complex number not in the spectrum of $T$. This relationship allows us to analyze operators' properties and facilitates functional calculus and spectral theory.
Resolvent Set: The resolvent set of an operator is the set of complex numbers for which the operator can be inverted, allowing the resolvent to be defined. This concept is crucial as it relates to the spectral properties of operators, influencing how they behave in various mathematical contexts, including spectral theory for unbounded operators and the generation of C0-semigroups.
Spectral Mapping Theorem: The spectral mapping theorem is a fundamental result in operator theory that describes how the spectrum of a bounded linear operator is related to the spectrum of a function applied to that operator. It connects the algebraic properties of operators and their spectral characteristics, particularly for holomorphic functions defined on the complex plane.
Spectrum: In operator theory, the spectrum of an operator refers to the set of values (complex numbers) for which the operator does not have a bounded inverse. It provides important insights into the behavior of the operator, revealing characteristics such as eigenvalues, stability, and compactness. Understanding the spectrum helps connect various concepts in functional analysis, particularly in relation to eigenvalues and the behavior of compact and self-adjoint operators.
Stability: Stability refers to the property of a system where the solutions to a dynamical equation remain bounded or converge to a steady state over time. In the context of generators of C0-semigroups, stability indicates that perturbations in the initial conditions or parameters do not lead to unbounded growth in the system's behavior, ensuring that the system behaves predictably and remains well-defined over time.
Strongly continuous: Strongly continuous refers to a property of a one-parameter semigroup of operators, where the semigroup is continuous with respect to the strong operator topology. This means that for any initial point in the space, as time progresses, the evolution of the system remains stable and converges to the expected limit, ensuring that the behavior of the operators does not exhibit abrupt changes. This continuity is crucial for understanding the dynamics of solutions to differential equations represented by these semigroups.
T(t): The notation t(t) typically represents the generator of a strongly continuous one-parameter semigroup, or C0-semigroup. It is a linear operator that describes the evolution of a dynamical system over time, linking the initial state of the system to its state at a later time through the semigroup operation. This concept is crucial in understanding the behavior of solutions to abstract differential equations, as well as in characterizing the properties of various operators associated with these systems.
Uniformly bounded: Uniformly bounded refers to a condition where a family of operators, or functions, are all bounded by the same constant across their entire domain. This concept is essential when discussing the properties of operators, especially in the context of semigroups, as it ensures that these operators do not exhibit uncontrolled growth and remain manageable in analysis.
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