Strongly steady half group

A strongly continuous semigroup (more precisely, strongly continuous operator semigroup , sometimes also referred to as a- semigroup ) is an object from the mathematical sub-area of functional analysis . Special cases of the strongly continuous semigroup are the norm-continuous semigroup and the analytic semigroup . ${\displaystyle C_{0}}$

definition

A family of continuous linear mappings of a real or complex Banach space in itself, which have the three properties ${\displaystyle \{T(t)\}_{t\geq 0}}$ ${\displaystyle T(t)\colon X\rightarrow X}$ ${\displaystyle X}$

1. ${\displaystyle T(0)=\operatorname {Id} }$,
2. ${\displaystyle T(s+t)=T(s)T(t)}$for everyone as well ${\displaystyle s,t\geq 0}$
3. ${\displaystyle \lim _{t\searrow 0}T(t)x=x}$ for all ${\displaystyle x\in X}$

fulfilled, is called strongly continuous semigroup . If you replace 3. with the stronger requirement

${\displaystyle \lim _{t\searrow 0}\|T(t)-\operatorname {Id} \|=0}$

so the family is called the norm-continuous half-group . ${\displaystyle \{T(t)\}_{t\geq 0}}$

If the semigroup can be continued holomorphically to a sector , then it is called analytical or holomorphic . ${\displaystyle \Sigma \subset \mathbb {C} }$

These semigroups play a major role in the (abstract) theory of evolution equations .

example

Let be a continuous linear operator, then define ${\displaystyle A\in L(X,X)}$

${\displaystyle T(t)=e^{tA}:=\sum _{n=0}^{\infty }{\frac {t^{n}A^{n}}{n!}}.}$

The series converges absolutely in and therefore defines a family of continuous linear operators. This family is a standardized semigroup and thus in particular also a strongly steady semigroup. ${\displaystyle L(X,X)}$

Classification of strongly continuous semigroups

For every strongly continuous semigroup there is one and one , so that the estimation for all ${\displaystyle \omega \in \mathbb {R} }$ ${\displaystyle M\geq 1}$ ${\displaystyle t\geq 0}$

${\displaystyle \|T(t)\|\leq M\mathrm {e} ^{\omega t}}$

applies. The operator norm on the Banach space denotes the continuous linear endomorphisms of . The semigroup is called ${\displaystyle \|\cdot \|}$ ${\displaystyle X}$

• as a contraction half-group , if this is fulfilled for and , ${\displaystyle M=1}$ ${\displaystyle \omega =0}$
• as a bounded semigroup , if the above inequality holds for one and , ${\displaystyle M\geq 1}$ ${\displaystyle \omega =0}$
• as a quasi-contractive semigroup , if the above inequality for and a is fulfilled. ${\displaystyle M=1}$ ${\displaystyle \omega \geq 0}$

The infimum above all possible , that is , is called the growth barrier . ${\displaystyle \omega _{0}}$ ${\displaystyle \omega }$ ${\displaystyle \omega _{0}:=\inf\{\omega \in \mathbb {R} \mid \exists \,M\geq 1\mathrm {\ mit\ } \|T(t)\|\leq M\mathrm {e} ^{\omega t},t\geq 0\}}$

If one considers instead , one speaks of strongly continuous groups . ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle t>0}$

Strongly steady semi-groups can under certain circumstances be continued from on sectors in the complex level. Such semigroups are called analytical . ${\displaystyle t\in [0,\infty )}$

Infinitesimal producer

Be a strongly continuous semigroup. The mapping is called an infinitesimal generator or an infinitesimal generator of ${\displaystyle \{T(t)\}_{t\geq 0}}$
${\displaystyle \{T(t)\}_{t\geq 0}}$

${\displaystyle A\colon \,{\mathcal {D}}(A)\to X,\,x\mapsto \lim _{t\searrow 0}{\frac {T(t)x-x}{t}}}$

with the domain of definition

${\displaystyle {\mathcal {D}}(A):=\left\{x\in X\;\left|\;\lim _{t\searrow 0}{\frac {T(t)x-x}{t}}{\text{ existiert}}\right.\right\}.}$

${\displaystyle A}$is a tightly defined, closed , linear operator.

${\displaystyle A}$is bounded if and only if it converges to the identity even in the operator norm . ${\displaystyle T(t)}$

The abstract Cauchy problem

${\displaystyle \left\{{\begin{array}{lll}u'(t)&=&A(u(t))+f(t)\ \mathrm {f{\ddot {u}}r\ alle} \ t\geq 0\ ,\\u(0)&=&u_{0}\\\end{array}}\right.}$

for the initial value and a continuously differentiable function is given by the function ${\displaystyle u_{0}\in {\mathcal {D}}(A)}$ ${\displaystyle f\colon [0,\infty )\rightarrow X}$

${\displaystyle u(t):=T(t)u_{0}+\int _{0}^{t}T(t-s)f(s){\rm {d}}s}$

solved.

The following applies to the spectrum of the producer: Is , then applies , where the growth barrier is the semigroup. ${\displaystyle \lambda \in \sigma (A)}$ ${\displaystyle \mathrm {Re} \,\lambda \leq \omega _{0}}$ ${\displaystyle \omega _{0}}$

The resolvent of coincides with the Laplace transformation of the semigroup to the right of the growth barrier, so it is true for and all . ${\displaystyle A}$ ${\displaystyle \textstyle R(\lambda ,A)x=\int _{0}^{\infty }\mathrm {e} ^{-\lambda t}T(t)x\mathrm {d} t}$ ${\displaystyle \mathrm {Re} \lambda >\omega _{0}}$ ${\displaystyle x\in X}$

Hille-Yosida's theorem

Of particular interest is whether a given operator is the infinitesimal generator of a strongly continuous semigroup. This question is fully answered by Hille - Yosida's theorem : ${\displaystyle A}$

A linear operator is then the infinitesimal generator of a strongly continuous semigroup which satisfies the estimate if it is closed and densely defined, a subset of the resolvent set of is and ${\displaystyle A\colon D(A)\subset X}$ ${\displaystyle \{T(t)\}_{t\geq 0}}$ ${\displaystyle \|T(t)\|\leq M\mathrm {e} ^{\omega t}}$ ${\displaystyle A}$ ${\displaystyle (\omega ,\infty )}$ ${\displaystyle A}$

${\displaystyle \left\|(\lambda I-A)^{-n}\right\|\leq {\frac {M}{(\lambda -\omega )^{n}}}}$for everyone and . ${\displaystyle \lambda \in (\omega ,\infty )}$ ${\displaystyle n\in \mathbb {N} }$

application

One application is that you want to solve the evolution equation with a given differential operator . Hille-Yosida's theorem says that one has to investigate the resolvent equation, which then leads to elliptical problems. If one can solve the elliptical problem, it is easy to solve the evolution problem. ${\displaystyle u'=Au+f}$ ${\displaystyle A}$

Derivation

The theory of the strongly continuous semigroups developed from the consideration of the Cauchy problem. The simplest form of the Cauchy problem is the question of whether a differentiable function exists for a given and an initial value , which ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle u_{0}\in \mathbb {R} }$ ${\displaystyle u\in C^{1}(\mathbb {R} )}$

${\displaystyle \left\{{\begin{array}{rcll}u'(t)&=&au(t)&\mathrm {f{\ddot {u}}r\ } t\geq 0\\u(0)&=&u_{0}\end{array}}\right.}$

Fulfills. From the theory of ordinary differential equations we get that is clearly given by . This can now be generalized by looking at the problem in higher dimensions, i.e. choosing an initial value and a matrix. Again, the solution is from ${\displaystyle u}$ ${\displaystyle u(t):=\mathrm {e} ^{at}u_{0}}$ ${\displaystyle u_{0}\in \mathbb {R} ^{n}}$ ${\displaystyle A}$ ${\displaystyle n\times n}$ ${\displaystyle u=\mathrm {e} ^{tA}u_{0}}$

${\displaystyle \left\{{\begin{array}{rcll}u'(t)&=&Au(t)&\mathrm {f{\ddot {u}}r\ } t\geq 0\\u(0)&=&u_{0}\end{array}}\right.}$.

Here the matrix exponential function is defined by , as in the real world . The Cauchy problem can also be posed on a Banach space in which and is chosen as an operator on . Is a bounded operator, then to turn the solution of the Cauchy problem. Operators that occur in the application, such as the Laplace operator, raise the question of generalizing to discontinuous operators, since in this case the sum does not generally converge. This gives rise to the problem of how to define the exponential function in the case of an unbounded operator. Independently of each other, Einar Hille and Kōsaku Yosida were able to present a solution around 1948: ${\displaystyle \textstyle \mathrm {e} ^{tA}:=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}A^{n}}$ ${\displaystyle X}$ ${\displaystyle u_{0}\in X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle u=\mathrm {e} ^{tA}u_{0}}$ ${\displaystyle \textstyle \mathrm {e} ^{tA}:=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}A^{n}}$ ${\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}A^{n}}$

Hille's approach: starting from the identity valid in the real , one obtains . This representation has the advantage that the resolvent is restricted and therefore only restricted operators appear on the right-hand side. Hille was able to show that under certain circumstances the limit of this sequence exists. If one considers a strongly continuous semigroup , as it is defined in the introduction, with its producer , it fulfills the equation . ${\displaystyle \textstyle \mathrm {e} ^{tA}=\lim _{n\rightarrow \infty }\left(1-{\frac {t}{n}}A\right)^{-n}}$ ${\displaystyle \textstyle \mathrm {e} ^{tA}=\lim _{n\rightarrow \infty }\left({\frac {n}{t}}R\left({\frac {n}{t}},A\right)\right)^{n}}$ ${\displaystyle T}$ ${\displaystyle A}$ ${\displaystyle \textstyle T(t)=\lim _{n\rightarrow \infty }\left({\frac {n}{t}}R\left({\frac {n}{t}},A\right)\right)^{n}}$

Yosida approximation : Yosida's idea was to define the (unlimited) operator by a sequence of limited operators. To do this, he continued and showed that converges in point against . Furthermore produce as bounded operators strongly continuous half groups with which for each point-in against an operator converge. The family of operators is in fact a strongly continuous semigroup, and every strongly continuous semigroup can be approximated by the Yosida approximation. ${\displaystyle A}$ ${\displaystyle A_{n}:=nAR(n,A)}$ ${\displaystyle A_{n}}$ ${\displaystyle D(A)}$ ${\displaystyle A}$ ${\displaystyle A_{n}}$ ${\displaystyle T_{n}}$ ${\displaystyle T_{n}(t)=\mathrm {e} ^{tA_{n}}}$ ${\displaystyle t\geq 0}$ ${\displaystyle X}$ ${\displaystyle T(t)}$ ${\displaystyle \{T(t)\}_{t\geq 0}}$

literature

• Klaus-Jochen Engel, Rainer Nagel : One-parameter semigroups for linear evolution equations. Springer, New York NY 2000, ISBN 0-387-98463-1 ( Graduate Texts in Mathematics 194).
• Einar Hille , Ralph S. Phillips : Functional Analysis and Semi-Groups. Revised and expanded edition. American Mathematical Society, Providence RI 2000, ISBN 0-8218-1031-6 ( American Mathematical Society. Colloquium publications 31).
• Tosio Kato : Perturbation Theory for Linear Operators. Corrected printing of the 2nd edition. Springer, Berlin 1980, ISBN 0-387-07558-5 ( The basic teachings of the mathematical sciences in individual representations 132), (Reprint. Springer-Verlag, Berlin et al. 1995, ISBN 3-540-58661-X ( Classics in mathematics )).
• Ammon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin et al. 1983, ISBN 3-540-90845-5 ( Applied Mathematical Sciences 44).