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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.

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Prove that $BE^{\alpha}$ is a Banach Space

Let $E$ a Banach space, can we proof that the set $$BE^{\alpha}=\{f \in S': \ \sup_{t>0}t^{\alpha}\|G(t)f\|_{E}\}$$ is also a Banach space? Here $S'$ is the set of tempered distributions $S'(\...
Jarbas Dantas Silva's user avatar
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About a property of solutions to the delay equation $\frac{dx}{dt} = Ax(t) + F(t,\,x_t)$ in a Banach space $E$

I'm dealing with the equation \begin{equation*} \tag{1} \frac{dx}{dt} = Ax(t) + F(t, \, x_t),\end{equation*} in which $A$ is the generator of a $C_0$-semigroup $(T(t))_{t \, \geqslant \, 0}$ on a ...
user405919's user avatar
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1 answer
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Exponential of nonlinear operator for a Cauchy problem

Does the exponential of a nonlinear operator solve the Cauchy problem for an ODE of say, this form \begin{align*} &\frac{dy}{dt}=f(t,y(t))\\ &y(0)=y_0 \end{align*} so is this true? \begin{...
Aner's user avatar
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Reference request: uniformly continuous semigroups of nonlinear (Lipschitz) operators

Consider a Banach space $X$ with norm $\vert\cdot\vert$, and call an operator $A\colon X\to X$ Lipschitz whenever $$\sup_{f\neq g} \frac{\vert Af-Ag\vert}{\vert f-g\vert}<+\infty;$$ the Lipschitz ...
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Are there any useful convexity properties of quantum dynamical semigroups?

I'm am wondering if there are any useful properties of quantum dynamical semigroups I can exploit for convex/concave optimization with respect to the semigroup parameter. A proper definition of a ...
nlupugla's user avatar
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1 answer
45 views

An application of the uniform boundedness principle to evaluate the norm.

I'm trying to show the following question: Suppose $\{T_t\}_{t\geq0}$ is a family of bounded linear operators over a Banach space $X$ satisfying $T_0=I, T_tT_s=T_{t+s}$. And for all $x \in X$, the map ...
Leven Wong's user avatar
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1 answer
53 views

Positivity of the bidual operator

How is the fixed space of a bounded linear operator related to the fixed space of the bidual operator? Motivation of question: I have a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach ...
Guest's user avatar
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Sectoriality of the 1-dimensional Laplace operator in $H^{-1}$

I know how to prove that the 1-dimensional Laplace operator is sectorial in $L^{2}$ with the domain being $H_{0}^{1}\cap H^{2}$. I've been wondering how to prove that it is also sectorial in $H^{-1}$ ...
AnonymousUser's user avatar
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1 answer
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Looking for an argument why $e^{-tK}$ is completely positive for $K(A):=\frac{1}{2}(VA+AV^*)$

A completely positive map $\Phi : M_d(\mathbb{C})\longrightarrow M_d(\mathbb{C})$ is a map such that for any $n \in \mathbb{N}$ the map $$\Phi \otimes Id_n : M_d(\mathbb{C})\otimes M_n(\mathbb{C}) \...
CoffeeArabica's user avatar
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The generator of analytic semigroup

Definition: Let $0<\omega<\frac{\pi}{2}$. A family of bounded operators $\{S(z):z\in S^0_{\omega_{+}}\}$ on $X$ is called a holomorphic semigroup if $S(z)S(\omega)=S(z+\omega) $ for all $z,\...
accretive's user avatar
4 votes
1 answer
126 views

What can semigroup theory do better in the study of PDEs compared to alternative methods?

I've recently come across semigroup theory in my mathematical physics class and while the theory itself feels nice to work with, I have not yet understood what does the theory offer for the study of ...
Cartesian Bear's user avatar
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15 views

Does there exist a convolution semigroup with compactly supported density?

I am looking for a convolution semigroup $(P_t)_{t \geq 0}$ of operators acting on $C_0^\infty(\mathbb{R^d})$, of the form $$P_tf(x) = (p_t * f)(x) = \int_{\mathbb{R}^d} p_t(x-y) f(y) dy,$$ where $p_t$...
Julius's user avatar
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Uniform Boundedness of a $C_0$-group

Let $p\in [1,\infty)$. On $X=L^p(\mathbb{R}; \mathbb{C}^2)$ we consider the operator $D=A+B$, where $A=\begin{pmatrix}-\partial_x & 0 \\ 0 & \partial_x \end{pmatrix}\quad{and}\quad B=c(x)\...
ym94's user avatar
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The definition of a continuous semigroup

Here is the definition of a continuous semigroup Let $C$ be a subset of a Banach space $X$. A semigroup on $C$ is a group $\{S(t):t\geq 0\}$ of a self-maps defined on the subset $C$ which satisfies ...
ran's user avatar
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does the regularized heat kernel have the semigroup property?

Consider the heat kernel on the 2D-torus $$ K(t,z-z_1):= \sum_{m \in \mathbb{Z}^2} \exp\Big(-(1-4\pi^2|m|^2)t +2\pi i m \cdot (z-z_1)\Big) $$ It satisfies the semigroup property $$ K(t_1+t_2,z_1-z_2)=\...
Marco's user avatar
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Analyticity of the semigroup on unimodular Lie group generated by the sublaplacian

Let $G$ a connected unimodular Lie group, endowed with Haar measure $X={X_1,\cdots,X_k}$ a Hörmander system of left-invariant vector fields. The sublaplacian $\Delta = - \sum_{i=1}^k X_i^2$ generates ...
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Heat semigroup is self-adjoint

Consider a closed Riemannian manifold $(M,g)$ and the heat equation $\partial_tu = \Delta_g u$ on it. Let $P_t$ be the heat semigroup generated by the equation. Of course the laplacian is symmetric ...
theflame's user avatar
4 votes
1 answer
64 views

Action of exponential of multiplication operator on $L^2$

Let $X: L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ be a multiplication operator, i.e. $f(x) \mapsto xf(x)$ for $f \in L^2$. Multiplication operators are known to be self-adjoint on some dense subset ...
CBBAM's user avatar
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2 votes
1 answer
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Do exponentials in semigroups only have formal meaning?

Let $L$ be a linear differential operator and consider the PDE $$\begin{cases} u_t + Lu = 0, \quad x \in \mathbb{R}^n\\ u = f, \quad t = 0\end{cases} \tag{1}$$ It is known that we may construct a ...
CBBAM's user avatar
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Characterization of continuous flows on an interval

I'm struggling with the proof of the following result: Proposition Let $-\infty \leq a < b \leq \infty$. A mapping $\phi: \mathbb{R} \times (a,b) \rightarrow (a,b)$ defines a continuous flow if ...
Scottish Questions's user avatar
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44 views

Is $C_0^{\infty}(\mathbb{R}^d)$ a core for diffusion operators on $C_0$?

This question is motivated by Hille-Yosida semigroup/Markov theory. Let $P_t:C_0(\mathbb{R}^d) \to C_0(\mathbb{R}^d), t \geq 0$ be a strongly continuous contraction semigroup. Say a function $f \in ...
Alex's user avatar
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1 answer
72 views

A question about Schrödinger semigroups

The question is about Theorem 14.1 in Teschl's Partial Differential Equations which goes like that $\textbf{Theorem 14.1.}$ The family $T_S(t)$ is a $C_0$-group in $H^{r}(\mathbb{R}^{n})$ whose ...
MathGeek's user avatar
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Generator of sub-Markov semigroup induces generator of Markov semigroup

I have to show that for the generator $A:L¹ \rightarrow L¹$ of a sub-Markov semigroup and a non-negativ $f_* \in L¹$ (with $L¹$ Set of Lebesgueintegratable functions) with $\int_{-\infty}^\infty f_* = ...
Mathhead123's user avatar
3 votes
1 answer
164 views

Given dense $A: X\to X$, is $A^2$ dense?

Let $A: X\to X$ be a densely defined linear operator acting on Banach Space $X$. Is $A^2$ also dense? Context: I was trying to think of an alternative proof for the Hille-Yosida theorem - ...
Kyan Cheung's user avatar
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Does an operator that "commutes" with the infinitesimal generator "commute" with the generated semigroup?

Consider two Markov semigroups $(T_t)_{t\geq 0}$ and $(S_t)_{t\geq 0}$ with infinitesimal generators $(A, D(A))$ and $(B, D(B))$. Suppose that some (bounded) operator $\Gamma$ is such that $$ A\Gamma =...
Mushu Nrek's user avatar
2 votes
1 answer
127 views

Exponential boundedness of a strongly continuous semigroup $(T_t)_{t>0}$.

Let $T = (T_t)_{t>0} $ be a strongly continuous semigroup (of bounded operators) on a Banach space $E$, i.e, $ \lim_{t \rightarrow z} \|T_tx- T_{z}x \|, \forall z >0, \forall x \in E $. Note ...
Jeffrey Jao's user avatar
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0 answers
48 views

Stationary distribution for a function of Markov process

Suppose $E$ is a locally compact Polish space, and $(X_t)_{t\ge 0}$ is a Markov process on $E$ with a Feller transition semi-group $P_t:C_b(E)\to C_b(E)$ with a stationary or even ergodic probability ...
JY0's user avatar
  • 113
3 votes
2 answers
221 views

Every bounded linear operator is an infinitesimal generator

I'm studying the theory of semigroups from Pazy's book. I'm struggling to understand a specific inequality in the proof of a theorem stating that every bounded linear operator is an infinitesimal ...
mathematico's user avatar
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0 answers
38 views

Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard

In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
monotone operator's user avatar
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0 answers
31 views

Semigroup solving Schrödinger equation weakly is a unitary group

Thank you for reading! Setup: Let $(\mathcal{H},\langle \cdot |\cdot\rangle)$ be a separable complex Hilbert space, and $A:D(A)\to \mathcal{H}$ be a densely defined symmetric unbounded operator in $\...
crimsonmist's user avatar
5 votes
0 answers
114 views

Existence of a weak-star limit of a family of operators in $B(\mathcal{H})$

Thank you in advance for reading this question, and your thoughts. I am working with a family of operators $(A_{s,\alpha})_{s\geq 0,\alpha>0}$ in the space $B(\mathcal{H})$ (the space of bounded ...
crimsonmist's user avatar
1 vote
0 answers
81 views

Existence of an equivalent norm on a Banach space

I am trying to formulated a lemma, because my main result demainds a change of norm. I have an operator $A\colon D(A)\subset X\to X$, $X$ Banach, and $(0,\infty)\subset \rho(A)$. This operator has the ...
Luiza Camile's user avatar
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0 answers
19 views

limit with integrated semi-group

Let $S(t)_{t\geq 0}$ be an integrated semigroup on a vector space $E$. Let $ 0\leq\alpha< 1$, $ x\in E$. I haven't been able to calculate this limit, but I expect it to be zero. $$ \lim_{\epsilon \...
Laouadi besma's user avatar
2 votes
1 answer
129 views

The Spectrum of the derivative operator in a specific Banach space

Consider the Banach space $X=\left\{u\in C^1([0,1]):\, u(0)=0\right\}$ and the subspace $D=\{u\in C^2([0,1]):\, u(0)=u(1)=u'(0)=0\}$, and the operator $A:D\longrightarrow X$ defined by $Au=u'$. I have ...
amine's user avatar
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3 votes
1 answer
55 views

Searching for Functions Exhibiting Semigroup Property for Energy-Efficient Neuronal Modeling

I am working on designing energy-efficient neurons for neuromorphic computing. One of the critical aspects of the dynamics I'm exploring is that they should adhere to the semigroup property. ...
Andi Faust's user avatar
3 votes
1 answer
49 views

Proving that $(G_\lambda)_\lambda$ is a resolvent family

Let $\mathcal{E}$ be a bilinear form on dense subset $\mathcal{D}\subset H$ of a Hilbert space $H$. Assume $\mathcal{E}$ is a closed, symmetric and positive definite, i.e., $\mathcal{E}(u,u)\geq0$. ...
Guy Fsone's user avatar
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58 views

The derivative of the Heat semigroup is the convolution by the derivative of the heat kernel?

Let $(T_t)_{t \geq 0}$ be the Heat semigroup of operators acting on the space $L^p(\mathbb{R}^n)$ with kernel $k_t : \mathbb{R}^n \to \mathbb{R}$. I know that that the map $t \mapsto T_t$ is ...
Liam's user avatar
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0 answers
24 views

Can we find an explicit solution of this Poisson equation?

Let $\kappa$ be a Markov kernel with invariant measure $\pi$ and $$A:=\kappa-\operatorname{id}$$ denote the corresponding (discrete-time) generator of $\kappa$. Let $c>0$ and $$r:=\frac cp$$ where $...
0xbadf00d's user avatar
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0 answers
46 views

How are the variance of a Markov chain ergodic average and the variance of the corresponding continuous time embedding related?

Let $(E,\mathcal E)$ be a measurable space; $\kappa$ be a Markov kernrel on $(E,\mathcal E)$; $\mu$ be a probability measure on $(E,\mathcal E)$ invariant with respect to $\kappa$ $A:=\kappa-\...
0xbadf00d's user avatar
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0 answers
39 views

References for regularity properties of diffusion/Markov semigroups

The question is related to a technical point in Bakry-Emery calculus (page 130 of Bakry, Gentil, and Ledoux's book "Analysis and geometry of Markov diffusion operators"): given a Markov ...
pencilwriter's user avatar
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0 answers
64 views

Expressing $\int_0^t\operatorname E_\nu\left[f(X_s)g(X_t)\right]\:{\rm d}s$ as an expectation with respect to another measure $\mu$ instead of $\nu$

In the setting below, I want to compute $$\int_0^t\operatorname E_\nu\left[f(X_s)g(X_t)\right]\:{\rm d}s\tag0.$$ Let $E$ be a $\mathbb R$-Banach space; $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $...
0xbadf00d's user avatar
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1 vote
3 answers
95 views

Perturbation of uniformly continuous semigroup

I think I can prove the following claim about perturbing semigroups, but this must be known somehow, and I am just reinventing the wheel. Do you have any suggestions for references that treat this, ...
Daniele Avitabile's user avatar
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0 answers
32 views

What is the importance of Banach space in the theory of semigroups of linear operators?

Pazy's book defines semigroups as follows. Let $X$ be a Banach space. A one parameter family $T(t)$, $0< t < \infty$, of bounded linear operators from $X$ into $X$ is a semigroup of bounded ...
Ilovemath's user avatar
  • 3,004
3 votes
1 answer
183 views

Semigroup properties of spectral fractional Laplacian

I need to check if the different fractional Laplacians have the same properties regarding the semigroups they generate. All I know now is in the case of the Restricted Fractional Laplacian: $(-\Delta)^...
Mathslover's user avatar
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0 answers
59 views

Solving "time evolution" partial differential eq using Lagrange shift operator

I was reading about the Weierstrass transform $(W[\cdot])$, and how it's related to the difussion equation in one dimension. It's relation is given by that $W[f]$ is the convolution with the Heat ...
Daniel Muñoz's user avatar
0 votes
1 answer
35 views

Smoothness of heat kernel on Lipschitz and polygon (cornered) domain

I'm wondering about the spatial smoothness of the heat kernel $K(t,x,x_0)$ on Lipschitz and polygon domains (or cornered domains). It's well known that $K(t,x,x_0)$ is smooth in $t$ for very general ...
celebi's user avatar
  • 81
0 votes
2 answers
86 views

Polynomial decay of any order implies exponential decay for operators norm

Context and Motivation Consider a Banach space $X$ and a linear bounded operator $T \in \mathcal{L}(X)$. We have established that for each non-negative integer $k$, there exists a constant $C_k$ ...
Scottish Questions's user avatar
1 vote
1 answer
106 views

Is a dissipative closed operator maximal dissipative?

This is a follow up to the following questions about m-dissipativity: Are m-dissipative operators closed? Are m-dissipative operators closed (II)? Is a dissipative closed operator $A:D(A)\subset X\...
Trevor3's user avatar
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0 answers
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Heat semigroup in Heisenberg groups properties

I'm trying to find the construction of the heat semigroup in Heisenberg groups, but I haven't found anything. In particular, I'm trying to find out if the semigroup in the Heisenberg group has the ...
Ilovemath's user avatar
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0 answers
89 views

Semigroup of heat equation: $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$

For any $\kappa>0$, we consider the Gaussian heat kernel $$ p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}}, \quad t>0, x \in {\mathbb R}^d. $$ Let $L^0 := L^0 (\...
Akira's user avatar
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