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.

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41 views

Density of $H^1$ functions with bounded gradient

I have been working on a PDE problem and I came across the need of working with functions with bounded gradient. However, since I am working with semigroups, I need density of the domain of the ...
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32 views

Weak limits in continuous convolution semigroups

A convolution semigroup is a collection of probability measures $(\mu_t)_{t \in I}$ on $\mathbb R^d$, where $I \subset [0,\infty)$, for which $\mu_s * \mu_t = \mu_{s+t}$. The convolution semigroup is ...
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28 views

The dual semigroup is equivalent in norm to its original semigroup

I would like to show the following inequality regarding the dual semigroup of a semigroup of linear operators (the one at the end of the image). The screenshot comes from the book One-paramter ...
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37 views

The heat semigroup represented as $\{e^{t\Delta}; t>0\}$

Set $\{G(t);t>0\}$ the semigroup defined as $G(t):L^2 \to L^2$ for every $t>0$, where $$G(t)u=g_t*u$$ and $g_t(x)=(4\pi t)^{-\frac{n}{2}}\exp(-|x|²/4t)$, $\forall x \in \mathbb{R}^n$. On the ...
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1answer
36 views

Is it possible to approximate some PDE semigroups by explicit methods?

I'm concerned with the numerical methods for the approximations of semigroup associated to following Cauchy problems (which typically involves unbounded operators): $\begin{equation} \begin{array}{ccc}...
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47 views

Asymptotic behaviour of a shifted sinus function

Consider the function $f(x)=\cos(x)-\cos((1+t)x)$ for $t>0$ on $x\in\mathbb{R}$. I would like to show the following claim which is obvious from looking at the graph of the function. Claim. If $0<...
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44 views

Generator of the semigroup associated with the integral operator by Gaussian kernel

Let $X=C_\infty(\mathbb R^n)$, which is the closure of the Schwartz class $\mathcal S(\mathbb R^n)$ in $L^\infty(\mathbb R^n)$. For each $t\geq 0$, we define $T(t)$ on $X$ by $$T(t)u=\begin{cases} G_t*...
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1answer
34 views

$\|S(t)\| \leq Me^{ct} \Rightarrow \|u\|_{C(0,T,H)} \leq \|u_0\|_H+\|f\|_{L^1(0,T,H)}$?

Let $A$ be the infinitesimal generator of a $C_0$-semigroup of contractions $(S(t))$ in a Hilbert space $H$ and $f\in L^1(0,T;H)$. We know that the mild solution of the problem \begin{equation} \begin{...
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1answer
60 views

Uniformly continuous semigroups

The ultimate goal is to show that every uniformly continuous semigroup on a Banach space must be of the form $(e^{tA})_{t\geq 0}$ for some bounded linear operator $A$. There are many books discussing ...
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1answer
80 views

Does the Bi-Laplacian generate an analytic semigroup?

It i well-known that the Laplace operator generate an analytic semigroup for example in $L^2(\Omega)$ on nice domains $\Omega \in \mathbb{R}^n$. Now does also the Bi-Laplacian generate an analytic ...
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71 views

Is there some operator generator of an analytical semigroup with eigenvalues such that $\sum_{k} \frac{1}{|\lambda_k|} < \infty$?

The following is pretty relevant for my research: Consider the Laplacian $\Delta$ in 1-D then under certain conditions it is well known that it is diagonalizable and its eigenvalues are $\lambda_k=-k^...
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37 views

Is this set weakly compact?

Let $M$ be a compact manifold and $\mathcal C^0(M) =\{f:M\to \mathbb R; f\ \text{is continuous}\}$. Suppose that $T:\mathcal C^0(M) \to \mathcal C^0(M)$ a bounded linear operator such that $T$ is a ...
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104 views

Is the following operator (related to the Dirac's delta) bounded?

Consider $\eta \colon [-1,0] \to \mathbb{R}$ of bounded variation and define the (delay) operator $\Phi \colon \mathcal{C}[-1,0] \to \mathbb{R}$ by $$\Phi f = \int_{-1}^{0} f d \eta$$ for $f \in \...
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73 views

Is the derivation of the adjoint of this operator correct?

Consider for $r>0$ the Hilbert space $H=\mathbb{R} \times L^{2}[-r,0]$ with scalar product $$\langle x,y \rangle=x_0 y_0 + \int_{-r}^0 x_1(\theta) y_1(\theta) d\theta $$ for every $x=(x_0,x_1), y=(...
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78 views

Applications of Hille-Yosida Theorem to Partial Differential Equations

In the book "Partial Differential Equations" by Evans we are trying to solve the initial/boundary problem \begin{cases} \partial_tu+Lu=0 , (x,t)\in U_T\\ u=0 , (x,t)\in \partial ...
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1answer
58 views

Is the following operator invertible?

Let $H$ Hilbert and $A\colon D(A) \subset H \to H$ close (but unbounded) linear operator with dense domain. Assume moreover that it is maximally dissipative, i.e. $\langle Ax,x\rangle \leq 0$ for all $...
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29 views

Does nabla operator generate a strongly continuous semigroup?

Semigroup theory usually considers second-order equations (heat / wave equations). I was wondering if the theory can be applied to the first-order equations. It seems to me that the nabla operator ...
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85 views

Is the following claim true in $=\mathbb{R}^{d} \times L^{2}$?

In my thesis I'm studying an article with which I'm having the following problem: let $E=\mathbb{R}^{d} \times L^{2}([0,+\infty))$ be the Hilbert space with norm $ \|\alpha\|^{2}:=|x|^{2}+\|z\|_{L^{2}}...
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46 views

Is this operator surjective?

Let $H$ Hilbert space, $A \colon D(A) \subset H \to H$ be a linear, closed, densely defined operator. In [Fabbri, Giorgio, Fausto Gozzi, and Andrzej Swiech. "Stochastic optimal control in ...
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1answer
49 views

Elliptic operators are dissipative?

It is well known that in general the Laplace operator is a dissipative operator, i.e. if you call $A$ its realization on $D(A)=H^2(\Omega) \cap H_0^1(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is a ...
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73 views

Infinitesimal Generator of semigroup for markov chain

In Wikipedia the definition of generator of a semigroup is given as follows For a Feller process $(X_t)_{t\ge0}$ with Feller Semigroup $T=(T_t)_{t\ge0}$ and state space $E$ we define the generator $(...
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1answer
34 views

Contractions semigroup

Let $H$ Hilbert, $\{e_k\}$ orthonormal basis, $A \colon D(A) \subset H \to H$ generator of a strongly continuous semigroup $e^{At}$ and $A$ such that $$Ae_k=-\lambda_k e_k$$ for some eigenvalues $\...
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26 views

Estimative for Heat semigroup : heat equation with a potential

Consider the equation: \begin{align*} u_t -\triangle u -a(t,x)u&= f(t,x),\\ u|_{\partial \Omega} &=0, \\ u(0)&=u_0. \end{align*} with $\Omega\subset \mathbb{R}^N$ open and $|\Omega|<\...
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1answer
46 views

Nonlinear semi group

Good Morning, I search for a theorem in the theory of a nonlinear semigroup, which gives the global existence of a solution of a PDE, with some regularity of course. Are there any recommended ...
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69 views

Help understanding exponential in Feynman-Kac proof

I am working through the proof of the Feynman-Kac formula in Johnson and Lapidus' book The Feynman Integral and Feynman's Operational Calculus. In one of the steps of the proof they write that $$e^{-t(...
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126 views

Motivation of the Proof of the Hille-Yosida Theorem

Let $X$ be a Banach space and $A$ be a linear map from a subspace of $X$ to $X$. The Hille-Yosida theorem gives a necessary and sufficient condition for $A$ to be an infinitesimal generator of a ...
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48 views

Duhamel's formula for a critical operator

I have an operator that has a non-trivial solution to the homogeneous equation. Let $\hat{e}_0$ be the critical eigenvector and C(t) be the corresponding amplitude of an operator $\frac{d}{dt} - \...
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1answer
30 views

If $A$ is local, and $x$ is a point of relative maximum for $f$, then $Af(x)\le 0$,(Markov transition functions)

I have to set up the quesiton with some definitions, so before I get to the question there will be some definitions. Markov transition function(definition) Let $E$ be a topological space and let $\...
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15 views

If is $P(S(4^{-k}))$ true so $P(S(t))$ is too true?

i am reading an article about Besov-Morrey spaces an there has the following sentense: "The semigroup propety of $e^{t\Delta}$ allows us to reduce easily to $t = 4^{−k}$ for $k\in \mathbb{Z}$&...
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11 views

Tensor product of simple semigroup

I found the following in a paper: Let $(T_t,t\geq 1)$ be the following semigroup $T_t = [e^{-t}+(1-e^{-t})P]^{\otimes n}$ where $P$ is a probability measure, we know that $$[aQ+(1-a)P]^{\otimes n} = \...
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29 views

a set spans space

X is BANACH space and $T(t)_{t\geq0}$ un C$_{0}$-semigroupe sur X. U(x)= $int \{ y\in X, \exists (tn)\nearrow \infty \ limT_{tn}(x)=y\}$ My question is how to prove That if $U(x)\neq \emptyset$ then ...
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33 views

Is the following function continuous on $X$ Hilbert

Let X Hilbert space and $A\colon D(A) \subset X \to X$ a generator of a strongly continuous semigroup. So $D(A)$ is dense and $A$ is closed. Let $A^*$ its adjoint (which is dense since $D(A)$ is dense)...
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35 views

Show that $\{ T(t)\}_{t \ge0}$ is $C_0$-Semigroup.

I have been studying the the evolution equations in the last two months. I am studying the proof of the (Hille-Yosida) theorem, which is (Let $A$ be a linear operator on a Banach space, then $A$ is ...
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50 views

About EXISTENCE for PDE

In doing optimal control of Parabolic PDE's we often have to solve a problem like this: $$\begin{cases} \dfrac{\partial y}{\partial t}-d\Delta y(t,x)=f(y(t,x),p(t,x)) & (t,x)\in (0,T)\times\Omega ...
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55 views

generator of semigroup of multiplication operators on $L^p$

Suppose $1 \leq p <\infty$. Let $(T_t)_{t \geq 0}$ be a strongly continuous semigroup of multiplication operators on $L^p(0,1)$ defined by $T_t(f)=m_t \times f$ where the function $m_t \colon [0,1] ...
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1answer
63 views

Example of semigroup of mappings

Want some examples of semigroup of mappings. Let $X$ be a Banach space and consider the collection of mappings $\tau =\{T_t:t\ge 0\}$. Define, $T_t:X \to X$ by $$T_t(x)=xe^t \text{ for all }x\in X.$$ ...
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22 views

On the semigroup of the diffusion equation.

Hello. I cannot understand the conclusion of the study of the diffusion equation in "one-parameter semigroups of positive operators". More specifically the part where it is said that $(T(t))...
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36 views

Showing that the generator of the semigroup $(e^{tA})_{t\in\mathbb{R}}$ is $A$.

Hello. I am studying semigroup theory. In "one-parameter semigroups of positive operators" it is stated that $A$ is the generator. Why? Given the definition, it should be $$Af=\lim_{h\to 0} ...
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1answer
85 views

If $S(t)$ is a $C_0$-semigroup, is $S(t-s)f(s)$ Bochner integrable?

Let $X$ be a Banach space and let $S(t)$, $t \geq 0$, be a $C_0$-semigroup on $X$. Assume that $f : [0,+\infty) \rightarrow X$ is Bochner integrable. Is $S(t-s)f(s)$ Bochner integrable on $[0,t]$ and ...
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45 views

Is the complete positivity true for ultraweak continuous functions for countable infinite dimensional Hilbert spaces

Consider an ultraweakly continuous, completely positive generator $L$ of a Quantum Markov Semigroup that is defined on the trace class operators $\mathcal{L}_1$ on a countable infinite dimensional ...
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1answer
28 views

Why is the following operator $A^* B \in L(X)$?

Let $X$ Hilbert space. Let $A \colon D(A) \subset X \to X$ be the generator of a strongly continuous semigroup of linear bounded operators $e^{At}$, so possibly $A$ is an unbounded operator. Call $A^*$...
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41 views

range of $(I-\Delta)^{-1}$ on bounded domains

Let $\Omega \subset \mathbb{R}^n$ with smooth boundary. Let $A=\Delta$ laplace operator on $\mathcal{D}(A)=H^{2}(\Omega) \cap H_{0}^{1}(\Omega)$. Then in [Li, Xungjing, and Jiongmin Yong. Optimal ...
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1answer
45 views

Uniform convergence of semigroups of linear operators

In [Li, Xungjing, and Jiongmin Yong. Optimal control theory for infinite dimensional systems.1995] at page 241 it is claimed that: \begin{equation} \lim _{s \downarrow t}\left|e^{A^{*}(s-t)} \frac{y_{...
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1answer
71 views

What does $\|\nabla e^{t\Delta}w\|_{L^p(\Omega)}$ mean?

I'm reading a paper that deals with the Neumann Heat Semigroup $e^{t\Delta}$ that until I know is defined like: $e^{t\Delta}=\displaystyle\sum_{k=0}^\infty \frac{(t \Delta)^k}{k!}$ what exactly mean $\...
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36 views

Relation between the Range(I+A) and Range(A)?

Let $H$ be a Hilbert space and consider $$A:D(A) \subset H \to H $$ to be unbounded linear monotone operator. Is there any relation between $\mathrm{Range}(I+A)$ and $\mathrm{Range}(A)$, generally? ...
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74 views

Is the "Dirichlet Laplacian" an extension of $\Delta$ on $C^2(Ω)$ or only on $\{u\in C(\overline Ω)\cap C^2(Ω):\left.u\right|_{\partial\Omega}=0\}$?

Let $\Omega\subseteq\mathbb R^d$ be bounded and open, $V:=H_0^1(\Omega)$ and $H:=L^2(\Omega)$. We know that there is a nondecreasing sequence $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $\...
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57 views

(left) Shift Semigroup and operator norm

I am reading some lecture notes on strongly continuous semigroups. I am having difficulty understanding an example: $$X = BUC(\mathbb{R}) := \{f:\mathbb R \rightarrow \mathbb R : f \text{ is uniformly ...
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2answers
215 views

A step in the proof of the Hille-Yosida theorem from Rudin

I'm getting stuck on perhaps a simple step in the Hille-Yosida theorem from 13.37 in Rudin's functional analysis. I wonder if someone has had this same difficulty before or knows how to get around it -...
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1answer
40 views

semigroup of operators generated by a diagonalizable operator and exponential

Let $A \colon D(A) \subset H \to H$ be the generator of a $C^0$ semigroup. Suppose that $e_k$ is an orthonormal basis and $A$ is diagonalizable $A e_k=\lambda_k e_k$ with eigenvalues $e_k$. Then is it ...
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13 views

Is there a unique relation between a time-homogeneous markov process and its semigroup?

I have been trying to find sources to verify this and I do sometimes find that Markov processes are defined through their transition semigroups, but I have not found explicitly that there is a unique ...

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