Strong steady group

A strongly continuous group is a family of bounded linear operators of a real or complex Banach space and is a special case of a strongly continuous semigroup . Strongly continuous groups are used in the investigation of partial differential equations that describe a reversible process . ${\displaystyle (T(t))_{t\in \mathbb {R} }}$ ${\displaystyle X}$

definition

Let be a Banach space and a family of bounded linear operators for . Applies ${\displaystyle X}$ ${\displaystyle T=(T(t))_{t\in \mathbb {R} }}$ ${\displaystyle T(t):X\rightarrow X}$ ${\displaystyle t\in \mathbb {R} }$

• ${\displaystyle T(0)=I}$,
• ${\displaystyle T(s+t)=T(s)T(t)}$for everyone and ${\displaystyle s,t\in \mathbb {R} }$
• ${\displaystyle \lim _{t\rightarrow 0}T(t)x=x}$for all , ${\displaystyle x\in X}$

this family is called the strongly steady group .

Infinitesimal producer

The (infinitesimal) generator is given by ${\displaystyle (A,D(A))}$

${\displaystyle D(A):=\left\{x\in X:\lim _{h\rightarrow 0}{\frac {T(h)x-x}{h}}\ \mathrm {existiert} \right\}}$

and

${\displaystyle Ax:=\lim _{h\rightarrow 0}{\frac {T(h)x-x}{h}}}$for . ${\displaystyle x\in D(A)}$

Inferences

• Create a strongly continuous semigroup with and a strongly continuous semigroup with a , and all . ${\displaystyle (A,D(A))}$ ${\displaystyle (T_{+}(t))_{t\geq 0}}$ ${\displaystyle \|T_{+}(t)\|\leq Me^{\omega t}}$ ${\displaystyle (-A,D(A))}$ ${\displaystyle (T_{-}(t))_{t\geq 0}}$ ${\displaystyle \|T_{-}(t)\|\leq Me^{\omega t}}$ ${\displaystyle \omega \in \mathbb {R} }$ ${\displaystyle M>0}$ ${\displaystyle t>0}$

So is the producer of a very steady group with for , for and for . ${\displaystyle (A,D(A))}$ ${\displaystyle (T(t))_{t\geq 0}}$ ${\displaystyle T(t)=T_{+}(t)}$ ${\displaystyle t\geq 0}$ ${\displaystyle T(t)=T_{-}(-t)}$ ${\displaystyle t<0}$ ${\displaystyle \|T(t)\|\leq Me^{\omega |t|}}$ ${\displaystyle t\in \mathbb {R} }$

• Be a tightly defined, closed operator and there exist and , such that and for all and all . ${\displaystyle (A,D(A))}$ ${\displaystyle \omega \in \mathbb {R} }$ ${\displaystyle M>0}$ ${\displaystyle (\omega ,\infty )\cup (-\infty ,-\omega )\subset \rho (A)}$ ${\displaystyle \|((|\lambda |-\omega )R(\lambda ,A))^{n}\|\leq M}$ ${\displaystyle \lambda >\omega ,\lambda <-\omega }$ ${\displaystyle n\in \mathbb {N} }$

Then create a strongly continuous group with for all . Here stand for the resolvent and for the resolvent quantity of . ${\displaystyle (A,D(A))}$ ${\displaystyle (T(t))_{t\in \mathbb {R} }}$ ${\displaystyle \|T(t)\|\leq Me^{\omega |t|}}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle R(\lambda ,A)}$ ${\displaystyle \rho (A)}$ ${\displaystyle A}$

Stone's theorem

Marshall Harvey Stone published the following sentence in the Annals of Mathematics in 1932 : Let be a Hilbert space and a strongly continuous group, where is unitary for all . Then there exists a self-adjoint operator such that is the generator of . Conversely, for every self-adjoint operator creates a strongly continuous group of unitary operators. ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle T(t)}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle iA}$ ${\displaystyle T}$ ${\displaystyle iA}$ ${\displaystyle A}$

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).
• 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).