It has been suggested that Quasicontraction semigroup be merged into this article. (Discuss) Proposed since November 2023. |
In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations.
Formally, a strongly continuous semigroup is a representation of the semigroup (R+, +) on some Banach space X that is continuous in the strong operator topology.
A strongly continuous semigroup on a Banach space is a map (where is the space of bounded operators on ) such that
The first two axioms are algebraic, and state that is a representation of the semigroup ; the last is topological, and states that the map is continuous in the strong operator topology.
The infinitesimal generatorA of a strongly continuous semigroup T is defined by
whenever the limit exists. The domain of A, D(A), is the set of x∈X for which this limit does exist; D(A) is a linear subspace and A is linear on this domain. [1] The operator A is closed, although not necessarily bounded, and the domain is dense in X. [2]
The strongly continuous semigroup T with generator A is often denoted by the symbol (or, equivalently, ). This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via functional calculus (for example, via the spectral theorem).
A uniformly continuous semigroup is a strongly continuous semigroup T such that
holds. In this case, the infinitesimal generator A of T is bounded and we have
and
Conversely, any bounded operator
is the infinitesimal generator of a uniformly continuous semigroup given by
Thus, a linear operator A is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator. [3] If X is a finite-dimensional Banach space, then any strongly continuous semigroup is a uniformly continuous semigroup. For a strongly continuous semigroup which is not a uniformly continuous semigroup the infinitesimal generator A is not bounded. In this case, does not need to converge.
Consider the Banach space endowed with the sup norm . Let be a continuous function with . The operator with domain is a closed densely defined operator and generates the multiplication semigroup where Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of can be derived by properties of . For example is bounded on if and only if is bounded. [4]
Let be the space of bounded, uniformly continuous functions on endowed with the sup norm. The (left) translation semigroup is given by .
Its generator is the derivative with domain . [5]
Consider the abstract Cauchy problem:
where A is a closed operator on a Banach space X and x∈X. There are two concepts of solution of this problem:
Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable. [6]
The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem: [7] Let A be a closed operator on a Banach space X. The following assertions are equivalent:
When these assertions hold, the solution of the Cauchy problem is given by u(t ) = T(t )x with T the strongly continuous semigroup generated by A.
In connection with Cauchy problems, usually a linear operator A is given and the question is whether this is the generator of a strongly continuous semigroup. Theorems which answer this question are called generation theorems. A complete characterization of operators that generate exponentially bounded strongly continuous semigroups is given by the Hille–Yosida theorem. Of more practical importance are however the much easier to verify conditions given by the Lumer–Phillips theorem.
The strongly continuous semigroup T is called uniformly continuous if the map t → T(t ) is continuous from [0, ∞) to L(X).
The generator of a uniformly continuous semigroup is a bounded operator.
A strongly continuous semigroup T is called eventually differentiable if there exists a t0 > 0 such that T(t0)X ⊂ D(A) (equivalently: T(t )X ⊂ D(A) for all t ≥ t0) and T is immediately differentiable if T(t )X ⊂ D(A) for all t > 0.
Every analytic semigroup is immediately differentiable.
An equivalent characterization in terms of Cauchy problems is the following: the strongly continuous semigroup generated by A is eventually differentiable if and only if there exists a t1 ≥ 0 such that for all x ∈ X the solution u of the abstract Cauchy problem is differentiable on (t1, ∞). The semigroup is immediately differentiable if t1 can be chosen to be zero.
A strongly continuous semigroup T is called eventually compact if there exists a t0 > 0 such that T(t0) is a compact operator (equivalently [8] if T(t ) is a compact operator for all t ≥ t0) . The semigroup is called immediately compact if T(t ) is a compact operator for all t > 0.
A strongly continuous semigroup is called eventually norm continuous if there exists a t0 ≥ 0 such that the map t → T(t ) is continuous from (t0, ∞) to L(X). The semigroup is called immediately norm continuous if t0 can be chosen to be zero.
Note that for an immediately norm continuous semigroup the map t → T(t ) may not be continuous in t = 0 (that would make the semigroup uniformly continuous).
Analytic semigroups, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous. [9]
The growth bound of a semigroup T is the constant
It is so called as this number is also the infimum of all real numbers ω such that there exists a constant M (≥ 1) with
for all t ≥ 0.
The following are equivalent: [10]
A semigroup that satisfies these equivalent conditions is called exponentially stable or uniformly stable (either of the first three of the above statements is taken as the definition in certain parts of the literature). That the Lp conditions are equivalent to exponential stability is called the Datko-Pazy theorem.
In case X is a Hilbert space there is another condition that is equivalent to exponential stability in terms of the resolvent operator of the generator: [11] all λ with positive real part belong to the resolvent set of A and the resolvent operator is uniformly bounded on the right half plane, i.e. (λI − A)−1 belongs to the Hardy space . This is called the Gearhart-Pruss theorem.
The spectral bound of an operator A is the constant
with the convention that s(A) = −∞ if the spectrum of A is empty.
The growth bound of a semigroup and the spectral bound of its generator are related by [12] s(A) ≤ ω0(T ). There are examples [13] where s(A) < ω0(T ). If s(A) = ω0(T ), then T is said to satisfy the spectral determined growth condition. Eventually norm-continuous semigroups satisfy the spectral determined growth condition. [14] This gives another equivalent characterization of exponential stability for these semigroups:
Note that eventually compact, eventually differentiable, analytic and uniformly continuous semigroups are eventually norm-continuous so that the spectral determined growth condition holds in particular for those semigroups.
A strongly continuous semigroup T is called strongly stable or asymptotically stable if for all x ∈ X: .
Exponential stability implies strong stability, but the converse is not generally true if X is infinite-dimensional (it is true for X finite-dimensional).
The following sufficient condition for strong stability is called the Arendt–Batty–Lyubich–Phong theorem: [15] [16] Assume that
Then T is strongly stable.
If X is reflexive then the conditions simplify: if T is bounded, A has no eigenvalues on the imaginary axis and the spectrum of A located on the imaginary axis is countable, then T is strongly stable.
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from into its bidual is a homeomorphism. A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this evaluation map is an isometric isomorphism and the normed space is a Banach space. Those space for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
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In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem.
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite dimension. It is one aspect of nonlinear functional analysis.
In functional analysis, the Hille–Yosida theorem characterizes the generators of strongly continuous one-parameter semigroups of linear operators on Banach spaces. It is sometimes stated for the special case of contraction semigroups, with the general case being called the Feller–Miyadera–Phillips theorem. The contraction semigroup case is widely used in the theory of Markov processes. In other scenarios, the closely related Lumer–Phillips theorem is often more useful in determining whether a given operator generates a strongly continuous contraction semigroup. The theorem is named after the mathematicians Einar Hille and Kōsaku Yosida who independently discovered the result around 1948.
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process.
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In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.
In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space. Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables is given a privileged position and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtained. Adding boundary conditions can often be translated in terms of considering solutions in some convenient function spaces.
This is a glossary for the terminology in a mathematical field of functional analysis.