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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.

2
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21 views

Given a Markov process, are we able to construct another Markov process with the same transition semigroup but different inital law?

Let $E$ be a locally compact separable metric space $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroups on $C_0(E)$ with generator $(\mathcal D(A),A)$ $(\Omega,\mathcal A,\operatorname P)...
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0answers
89 views
+50

How can we proof this implication in Corollary 4.8.7 in the book of Ethier and Kurtz?

I'm trying to understand the proof of the implication "(g) $\Rightarrow$ (f)" in Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas ...
0
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1answer
32 views

$L^p$-contractivity implies $L^p$-dissipativity?

Does $L^p$-contractivity of an operator semigrop imply the $L^p$-dissipativity of the operator? Please note the definition of $L^p$-dissipativity: $(Au, |u|^{p-2}u)\leq 0$ for all $u\in C^1_0(\...
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0answers
8 views

A question about how Positivity preserving property of heat kernel is used.

Sry this the second question from the following article, I am asking in this week. At page 6 (126), 3th line, of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say ...
0
votes
1answer
18 views

Iteration of resolvent operator

I am studying a paper about Markov semigroup and found an equation about resolvent operator. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, and $P(t):\mathcal{M}(\mathbb{R} )\...
2
votes
1answer
105 views

I don't understand the proof of Corollary 4.8.7 in the book of Ethier and Kurtz

I'm trying to understand the proof of Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas G. Kurtz. Here is the theorem and its proof:...
0
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0answers
16 views

Is the generator of a uniformly continuous contraction semigroup contractive?

Let $E$ be a $\mathbb R$-Banach space and $A$ be the generator of a uniformly continuous contraction semigroup $(T(t))_{t\ge0}$ on $E$. Are we able to derive some bound on $\left\|A\right\|_{\mathfrak ...
2
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1answer
37 views

Question about $e^{\frac{-itA}{\hbar}}(\hat{Q}+ \hat{P})e^{\frac{itA}{\hbar}}$

This arises in the context of trying to rigorously understand quantum dynamics but it's a functional analysis issue. For simplicity suppose we are in dimension $1$. Let $\hat{Q}$ be the operator ...
1
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0answers
27 views

Meaning of Compactness

Let $\Omega \subset\mathbb{R}$ be a bounded domain (interval) and observe the following problem : \begin{align*} (P) \begin{cases} u_{t} = \Delta u + |u|^{p-1}u\, \quad x\in\Omega, t>0\\ u(0,x) = ...
0
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1answer
21 views

Use of L'Hosptial Rule in generalized setting related to heat equation

Let $X$ be a space of continuous functions with compact support in a bounded domain $\Omega \subset \mathbb{R}^{N}$ with Lipschitz continuous boundary, $F : X \to X$ be a Lipschitz continuous function,...
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0answers
19 views

Can norm of resolvent of closed operator grow away from its spectrum(but near to its numerical range)

I hope my question is simple to answer but I could not find anything up to now. Given a closed densely defined operator $A \colon H \supseteq D(A) \to H$ in a Hilbert space $H$. For many types of ...
2
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0answers
23 views

Does the carré du champ operator associated with the generator of a contractive $C^0$-semigroup on a Hilbert space have the diffusion property?

Let $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on a $\mathbb R$-Hilbert space $H$ with generator $(\mathcal D(A),A)$, $\mathcal A$ be a subspace of $\mathcal D(A)$ with $fg\in\...
0
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1answer
32 views

If $T(t)$ is an immediately differentiable semigroup on $H$ with generator $A$, does $\frac{d}{dt}\|T(t)x\|_H^2=2⟨AT(t)x,T(t)x⟩_H$ hold for all $x∈H$?

Let $(T(t))_{t\ge0}$ be a semigroup on a $\mathbb R$-Hilbert space $H$ with $$\sup_{s\in[0,\:t)}\left\|T(s)\right\|_{\mathfrak L(H)}<\infty\tag1$$ for some (and hence all) $t>0$ and $(\mathcal D(...
1
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1answer
27 views

Show that the generator of a strongly continuous contraction semigroup on $L^2$ is nonpositive definite

Let $(E,\mathcal E,\mu)$ be a finite measure space, $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $L^2(\mu)$ and $(\mathcal D(A,A)$ denote the generator of $(T(t))_{t\ge0}$. ...
2
votes
1answer
70 views

If $T(t)$ is a semigroup on $E$ and $F$ is a subspace of $E$ such that $T(t)$ is $F$-preserving, how are the generators on $E$ and $F$ related?

Let $E$ be a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ be a semigroup on $E$ and $(\mathcal D(A),A)$ denote the generator of $(T(t))_{t\ge0}$. If $F$ is a closed subspace of $E$ and $T(t)F\...
0
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1answer
38 views

Heat semigroup representation

It is known that the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega),$ (with $\Omega$ being a open subset of $R^n$) generates a $C_0-$semigroup in $L^2(\Omega)$). Moreover, in ...
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0answers
34 views

Generation theorem for Feller semigroups

Let $E$ be a locally compact Hausdorff space. I want to show that a linear operator $(\mathcal D(A),A)$ on $C_0(E)$$^1$ is closable and the closure $(\mathcal D(\overline A),\overline A)$ is the ...
2
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1answer
55 views

How are the two versions of the Lumer-Phillips theorem given in the books of Engel/Nagel and Pazy related?

Let $E$ be a $\mathbb R$-Banach space and $(\mathcal D(A),A)$ be a densely-defined dissipative linear operator on $E$. In the book of Engel and Nagel, I've found the following verison of the Lumer-...
2
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0answers
73 views

If $\kappa$ is a contractive operator on $C_0(ℝ)$, is $λ(κ-\text{id})$ the generator of a Feller semigroup?

Let $E$ be a locally compact separable metric space, $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\},$$ $\kappa$ be a Markov kernel on $(...
8
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1answer
77 views

Heat semigroup norm between fractional Sobolev and $L^p$ spaces

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2\alpha,p}$ and classical Lebesgue space $L^q$? I am trying to derive an inequality $$ \lvert\...
0
votes
1answer
52 views

How can point spectrum equal residual spectrum?

I came across a result in a paper(Proposition 2.2 on the second page in https://www.ams.org/journals/tran/1988-306-02/S0002-9947-1988-0933321-3/S0002-9947-1988-0933321-3.pdf) which states: If $X$ ...
3
votes
0answers
66 views

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 $A^1$ $:...
1
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1answer
44 views

Exponential stability of a second order ordinary linear operator

Let's consider an unbounded second order linear differential operator $A := k(x)\frac{d^{2}}{dx^{2}}+\frac{d}{dx}$ defined over $L^{2}(0,1)$ whose domain is $H^{2}(0,1) \cap H_{0}^{1}(0,1)$. $k(x)$ is ...
5
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0answers
71 views

Is the generator of a semigroup of bounded linear operators closed even when the semigroup is not strongly continuous?

If $E$ is a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ is a semigroup of bounded linear operators on $E$ and $(\mathcal D(A),A)$ denotes the generator of $(T(t))_{t\ge0}$, is $(\mathcal D(A),A)$ ...
2
votes
1answer
23 views

Extension of domain of contactive semigroups, and why contractivity is important?

I am reading a paper, where I think the authors use this fact that the contractivity of a semigroup on a space of initial values, imeadiatly implies that we can extend the domain of semigroup to this ...
2
votes
1answer
32 views

Showing that $\lambda - (A + B)$ has dense range

Let $A$ be the generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$ of contractions on a Banach space $X$ and $B \in \mathcal L(X)$ a bounded operator. To apply some approximation formula I want to show ...
0
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1answer
31 views

Do self adjoint operators on a Hilbert space generates an analytic semigroup?

Is this generally true that, a densely defined, closed and self adjoint operator on a Hilbert space generates an analytic semigroup?
2
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2answers
58 views

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, \...
1
vote
1answer
106 views

Extending the domain of the Dirichlet form associated with a symmetric Markov semigroup

Let $(E,\mathcal E)$ be a measurable space $\mathcal M_b(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(P_t)_{t\ge0}$ be a Markov semigroup ...
0
votes
1answer
48 views

Generator of semigroup?

I have seen such expression: the generator of the semigroup is defined by $$ [Uf] = \lim_{t\rightarrow 0} \frac{U^t f - f}{t}. $$ However, I don't understand, isn't the $[Uf] = df/dt$ that simple?
4
votes
2answers
148 views

Show that the carré du champ operator is nonnegative

Let $(E,\mathcal E)$ be a measurable space $\mathcal M_b(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(\kappa_t)_{t\ge0}$ be a Markov ...
1
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0answers
76 views

Show that $C_c^∞(\mathbb R)$ is a core of the generator of the Feller semigroup induced by the strong solution of an SDE with Lipschitz coefficients

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ $W$ be ...
2
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0answers
49 views

$u'\in L^{\infty}(0,T;L^{2}(U))\cap L^{2}(0,T;H_{0}^{1}(U))$ regularity in the semigroup approach

Evans' PDE book 2nd, (Theorem 7.1.5) says for parabolic PDEs, if the initial condition is in $H^1_0 \cap H^2$, and the source term satisfies the regularity $\mathbf { f }, \mathbf { f } ^ { \prime } ...
1
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1answer
67 views

How can we prove that a (locally bounded) semigroup is strongly continuous on the closure of its generator?

Let $E$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a semigroup on $E$, i.e. $T(t)$ is a bounded linear operator on $E$ for all $t\ge0$, $T(0)=\operatorname{id}_E$ and $$T(s+t)=T(s)T(t)\;\;\;...
1
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1answer
11 views

Cesaro average semigroup

Let $(P_t)_{t\geq 0}$ be a symmetric strongly continuous semigroup on $L^2(X,\mu)$ with generator $(L,\mathcal{D}(L))$. Given $f\in\mathcal{D}(L^*)$, define the Cesaro average $$ A_tf=\frac{1}{t}\...
2
votes
1answer
32 views

Operator Semigroups Simplified

How do I explain Operator Semigroups, in particular, positive operator semigroups to someone who hasn't studied math beyond high school? I just want to give a vague idea/analogy to someone to let ...
0
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1answer
31 views

If $(T_t)$ is a semi group, why $\lim_{s\to 0}T_t\frac{T_su-u}{s}=T_t\lim_{s\to 0}\frac{T_su-u}{s}$?

Let $(T_t)_t$ is a continuous semi group (i.e. $\lim_{t\to 0}T_tu=u$ for all $u$). $T_t$ is a contraction for all $t$. We suppose that $\lim_{t\to 0}\frac{T_tu-u}{t}$ exist. Why $$\lim_{s\to 0}T_t\...
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0answers
39 views

self-adjoint bounded generates analytic semigroup

Engel Nagel A Short Course on Operator Semigroups Corollary II.4.8 states: (There should be a typo. If $\delta=0$ then the spectum is empty, but normal operator has a non-empty spectrum? Anyways,) ...
2
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1answer
61 views

Sectorial operator: $0\in \rho(A)$ or $0\not\in \rho(A)$?

I am confused with a characterisation of the infinitesimal generators that generates analytic semigroups. In the following characterisation do we or do we not need the origin (or if the sector is ...
3
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0answers
93 views

How is integration by parts applied here?

Let $C_0(\mathbb R)$ denote the space of continuous functions vanishing at infinity equipped with the supremum norm $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous, $$Lf:=bf'+\frac12\sigma^...
0
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1answer
47 views

Proof that bounded right continuous functions are integrable.

I am reading Davie's book "One parameter Semigroups", on page 16 in the proof that "weak semigroups" are also "strong semigroups" it claims that for a right continuous locally bounded function $g:\...
2
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0answers
35 views

How to prove that a linear differential operator generates a semigroup?

I am working on a nonlinear PDE and I should now prove that the linear operators $L:=\partial_x^3+\partial_x^2$, $S(u):=u L$ and $T(u):=L+S(u)$ are generators of a $C_0$ semigroup such that $\|e^{-s A}...
0
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1answer
31 views

Does strong convergence of generators imply the strong convergence of the corresponding semigroups?

Let $E$ be a $\mathbb R$-Banach space $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be $C^0$-semigroups on $E$ with generator $(\mathcal D(A_n),A_n)$ and $(\mathcal D(A),A)$, respectively, for $n\in\mathbb ...
0
votes
0answers
47 views

Show that a semigroup is strongly continuous on the domain of its generator

Let $(\kappa_t)_{t\ge0}$ be a semigroup of linear, conservative, contractive and nonnegative operators on $C_0(E)$ with $$(\kappa_tf)(x)\xrightarrow{t\to0}f(x)\tag1\;\;\;\text{for all }x\in\mathbb R$$ ...
0
votes
0answers
15 views

Restriction of generator of a semigroup

If an operator $A$ generates a semigroup on $L^2(\Omega)$, can we say that its restriction on $L^\infty(\Omega)$ is m-dissipatif ? Thank you.
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0answers
19 views

Proof that for a strongly continuous contraction resolvent, there is exactly one linear operator that generates the resolvent.

I have questions about the proof of the following Proposition from the book Introduction to the Theory of Non-Symmetric Dirichlet forms. First, how do we get the independence of $G_\alpha (B)$ of $\...
0
votes
0answers
36 views

What's exactly meant if one is saying that the closure of a multivalued operator is the generator of a $C^0$-semigroup?

Let $E$ be $\mathbb R$-Banach space $A$ be a subspace of $E\times E$ and $$\mathcal D(A):=\left\{x\in E\mid\exists y\in E:(x,y)\in A\right\}$$ $(T(t))_{t\ge0}$ be a contractive $C^0$-semigroup on $\...
2
votes
1answer
38 views

$C([0,T],(L(X),\mathcal T_{\text{strong}})) = C([0,T],(L(X),\mathcal T_{\text{uniform}}))$?

Given a Banach space $X$, we have two topologies on the space of all bounded linear operators $L(X)$, one is uniform operator topology $\mathcal T_{\text{strong}}$, the other is strong operator ...
1
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0answers
20 views

Parabolic semilinear Problems and Continuous Dependence on Initial Data

Let $(X, ||\,\cdot\,||)$ be a Banach Space and its norm. We want to look for $T>0$ and $u$ a solution of the following problem : $\begin{cases} u \in C([0,T],D(A))\cap C^{1}([0,T],X) \\ u'(t) = Au(...
1
vote
1answer
38 views

$(\int_0^\infty T(t) dt) f(x) = (\int_0^\infty T(t)f dt)(x) = \int_0^\infty T(t)f(x) dt$?

Given a sequence of bounded operators $\{T(t)\}_{t\ge0}$ defined on the Banach space $C_0(\mathbb R^d)$ equipped with the supremum norm $|f|_0:=\sup_{x\in\mathbb R^d}|f(x)|$. Suppose $$\int_0^\infty \|...