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Questions tagged [semigroup-of-operators]

For questions related to theory of semigroups of linear operators and its applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.

A semigroup is an algebraic structure consisting of a set together with an associative binary operation. This tag is used for questions related to theory of semigroups of linear operators and its applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.

636 questions
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Power of the infinitesimal generator

Let $A$ be the infinitesimal generator of a $C_0$ semigroup of linear operators in a Banach space. Let $n$ be a positive integer $n \geq 2$? Is the power operator $A^n$ closed? Here (setting $A^1$ $:=$ $A$, and denoting the domain of $A$ by…
Ma Pa
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Do contraction semigroups admit exponential representation?

Given a Banach space $\mathcal N$, as contraction semigroup is defined as a set of bounded operators $P^t$, $0\le t\le+\infty$ defined everywhere in $\mathcal N$, such that \begin{equation*} P^0=1, \hspace{5mm} P^tP^s=P^{t+s}, \hspace{5mm} t\ge 0,…
Graz
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Evolution Equation

Let $X=L^2(0,\pi)$. Define the operator $(A,D(A)$ by: $$D(A)=\{u\in H^2(0,\pi):u(0)=u'(\pi)=0\} ,\quad \quad Au=u''$$ Show that $A$ is the infinitesimal generator of a $C_0$ semigroup of contractions on $X$.
OKPALA MMADUABUCHI
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Translation semigroups

Consider the operators $Q(t)\in \mathcal{B}(L^{2}(\mathbb{R}))$ given by $(Q(t)f)(s)=f(s+t)\quad \forall f\in L^{2}(\mathbb{R})$. Each $Q(t)$ is clearly unitary and the following are satisfied: $Q(0)=$Identity on…
Arundhathi
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If $(T_t)_t$ is a strong continuous semigroup, why $T_sAf_n\to T_sg$ uniformly?

Let $H$ a Hilbert space and $(T_t)_{t>0}$ a strongly continuous semigroup. Let $(A,\mathcal D(A))$ its generator. Let $(f_n)$ a sequence of $\mathcal D(A)$ that converges to $f$ s.t. $(Af_n)_n$ converges to a function $g$. I want to show that…
user657324
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Analytic semigroup

Let $A$ be the infinitesimal generator of analytic semigroup $S(t)$ on a Hilbert space such that: $$\|S(t)\|\le \frac{M}{t^{\gamma}}$$ what we get fot $$\|AS(t)\|\le ??$$ I really appreciate any help you can provide.
Soufiane
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Domain of the infinitesimal generator of the shift semigroup

In a course at my university, we study strongly continuous semigroups and their infinitesimal generators. In a simple example, we take a look at a shift semigroup. let $ T $ be an operator on $ X = L^2(0, 1) $, defined by: if $ t + x \leq 1 $, $…
Ruben
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Semigroup not strongly continuous in 0

The solution of homogenous heat equation in bounded regular domain $\omega$ of $R^{n}$ is $$u(t,x)=\sum_{n\geq 1}a_{n}(0)\exp(-\lambda_{n}t)e_{n}=S(t) u_{0}$$ where $e_{n}$ is Hilbert basis of $L^{2}(\omega)$ and so $S(t)$ is a semigroup not…
Ama NI
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2 answers

Solver Equation

If $A : D(A) \subset X \rightarrow X$ is a closed operator, then \begin{eqnarray*} R(\lambda : A) - R(\mu : A) = (\mu - \lambda) R(\lambda : A)R(\mu : A), \ \ \forall \mu , \lambda \in \rho(A). \end{eqnarray*}
user253963
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$C_0$ Semigroups with same infinitesimal generator

I was dealing with this: Let $S(t)$ and $T(t)$ be $C_0$-semigroups with infinitesimal generators $A:D(A) \subset X \rightarrow X$ and $B:D(B) \subset X \rightarrow X$ respectively. Show that \begin{equation} A = B \hspace{5mm} \Rightarrow…
James Arten
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Strongly Continuous Semigroup consequence

I started studying theory of Semigroups recently and I was trying to figure out basic concepts. I know that A semigroup $S(t)$ of bounded linear operators on $X$ is called strongly continuous ($\mathcal(C_0)$ if \begin{equation} \displaystyle…
James Arten
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Conditions such that $T(t)x \in D(A)$ for $C_0$-Semigroup

Consider a $C_0$-Semigroup $(T(t))_{t \ge 0}$ on the Banachspace $X$. Are there any conditions on $x \in X$ or the semigroup such that for any $t > 0$ it holds $T(t)x \in D(A)$? Here $(A,D(A))$ denotes the Generator with its domain $D(A)$.
user317721
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Every $C_0$-Semigroup on $L^\infty$ admits a bounded Generator

I remember my professor telling us a statemen like: Every $C_0$-Semigroup on $L^\infty$ admits a bounded Generator. I currently need a result o this form, so i wonder if this ture. I couldn't find anything about this neither on google nor in my…
user317721
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0 answers

Estimation for $C_0$-semigroup

If we have some $C_0$-semigroup $(S(t))_{t\geq 0}$ and $\lVert u(t)\rVert\leq C$ for $t\in [0,T)$ and $u(0)=u_0$, why do we then have, for $0
Rhjg
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Semigroup generation

For an operator $(A,\mathcal{D}(A))$ on a Banach space $X$ define on $\mathcal{X} := X \times X$ the operator matrix $$\mathcal{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$$ with domain $\mathcal{D}(\mathcal{A}) :=…
sana
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