C₀-semigroup

Consider the Banach space ${\displaystyle C_{0}(\mathbb {R} ):=\{f:\mathbb {R} \rightarrow \mathbb {C} {\text{ continuous}}:\forall \epsilon >0~\exists c>0{\text{ such that }}\vert f(x)\vert \leq \epsilon ~\forall x\in \mathbb {R} \setminus [-c,c]\}}$ endowed with the sup norm ${\displaystyle \Vert f\Vert$ :={\text{sup}}_{x\in \mathbb {R} }\vert f(x)\vert } . Let ${\displaystyle q:\mathbb {R} \rightarrow \mathbb {C} }$ be a continuous function with ${\displaystyle {\text{sup}}_{s\in \mathbb {R} }{\text{Re}}(q(s))<\infty }$. The operator ${\displaystyle M_{q}f:=q\cdot f}$ with domain ${\displaystyle D(M_{q}):=\{f\in C_{0}(\mathbb {R} ):q\cdot f\in C_{0}(\mathbb {R} )\}}$ is a closed densely defined operator and generates the multiplication semigroup ${\displaystyle (T_{q}(t))_{t\geq 0}}$ where ${\displaystyle T_{q}(t)f:=\mathrm {e} ^{qt}f.}$ Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of ${\displaystyle M_{q}}$ can be derived by properties of ${\displaystyle q}$. For example ${\displaystyle M_{q}}$ is bounded on ${\displaystyle C_{0}(\mathbb {R)} }$ if and only if ${\displaystyle q}$ is bounded.[4]

Let ${\displaystyle C_{ub}(\mathbb {R} )}$ be the space of bounded, uniformly continuous functions on ${\displaystyle \mathbb {R} }$ endowed with the sup norm. The (left) translation semigroup ${\displaystyle (T_{l}(t))_{t\geq 0}}$ is given by ${\displaystyle T_{l}(t)f(s):=f(s+t),\quad s,t\in \mathbb {R} }$.

Its generator is the derivative ${\displaystyle Af:=f'}$ with domain ${\displaystyle D(A):=\{f\in C_{ub}(\mathbb {R} ):f{\text{ differentiable with }}f'\in C_{ub}(\mathbb {R} )\}}$.[5]

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

${\displaystyle \omega _{0}=\inf _{t>0}{\frac {1}{t}}\log \|T(t)\|.}$

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

${\displaystyle \|T(t)\|\leq Me^{\omega t}}$

for all t ≥ 0.

The following are equivalent:[10]

1. There exist M,ω>0 such that for all t ≥ 0: ${\displaystyle \|T(t)\|\leq M{\rm {e}}^{-\omega t},}$
2. The growth bound is negative: ω₀ < 0,
3. The semigroup converges to zero in the uniform operator topology: ${\displaystyle \lim _{t\to \infty }\|T(t)\|=0}$,
4. There exists a t₀ > 0 such that ${\displaystyle \|T(t_{0})\|<1}$,
5. There exists a t₁ > 0 such that the spectral radius of T(t₁) is strictly smaller than 1,
6. There exists a p ∈ [1, ∞) such that for all x ∈ X: ${\displaystyle \int _{0}^{\infty }\|T(t)x\|^{p}\,dt<\infty }$,
7. For all p ∈ [1, ∞) and all x ∈ X: ${\displaystyle \int _{0}^{\infty }\|T(t)x\|^{p}\,dt<\infty .}$

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 L^p 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 ${\displaystyle H^{\infty }(\mathbb {C} _{+};L(X))}$. This is called the Gearhart-Pruss theorem.

The spectral bound of an operator A is the constant

${\displaystyle s(A):=\sup\{{\rm {Re}}\,\lambda$ :\lambda \in \sigma (A)\}} ,

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:

• An eventually norm-continuous semigroup is exponentially stable if and only if s(A) < 0.

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: ${\displaystyle \lim _{t\to \infty }\|T(t)x\|=0}$.

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

1. T is bounded: there exists a M ≥ 1 such that ${\displaystyle \|T(t)\|\leq M}$,
2. A has not residual spectrum on the imaginary axis, and
3. The spectrum of A located on the imaginary axis is countable.

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.