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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.

13
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1answer
2k views

Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very similar ...
9
votes
2answers
444 views

Semi-group theory and Poisson equation on the upper half plane

We first look at the 2D Laplace equation , say on the upper half plane: $$\Delta u=0,\quad -\infty<x<\infty, y>0$$ $$u(x,0)=g(x),$$ where $g\in L^p(\mathbb{R})$ for some $1\leq p<\infty$. ...
8
votes
1answer
2k views

What does Trotter Product Formula mean?

For some reason, I have to work with Trotter product formula recently, but I do not have a strong background in functional analysis. The following is the statement of the formula from MathWorld ...
8
votes
1answer
687 views

Does Continuity in Weak Operator Topology imply Continuity in Strong Operator Topology?

This homework problem has puzzled me for almost a year. As nobody in the class has figured it out, I would like to seek a proof or a disproof here. Problem (Prove or Disprove) Let $X$ be a Banach ...
8
votes
1answer
576 views

Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...
8
votes
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\...
7
votes
2answers
363 views

Continuity of semigroups on $L^2$ and $L^1$: Is this simple proof correct?

Let $(X, \mu)$ be a $\sigma$-finite measure space, and $P_t$ a symmetric, Markovian, strongly continuous contraction semigroup on $L^2(X,\mu)$. (Markovian means that if $f \in L^2$ with $0 \le f \le ...
7
votes
2answers
2k views

Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in X$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function $U:[...
7
votes
1answer
738 views

Weak generator of Feller semigroup

Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in \mathbb{...
7
votes
1answer
165 views
+100

If $X^{(n)},X$ are càdlàg and $X^{(n)}\to X$ in distribution, do the corresponding transition semigroups strongly converge?

Let $\left(\kappa^{(n)}_t\right)_{t\ge0}$ and $(\kappa_t)_{t\ge0}$ be Markov semigroups on $(\mathbb R,\mathcal B(\mathbb R))$ for $n\in\mathbb N$ $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly ...
6
votes
1answer
657 views

Show that the semigroup S(t) here described is a contraction semigroup

Definition 1: Let $H$ be a Hilbert space. A strongly continuous semigroup is a family $\{S(t)\}_{t \ge 0}$ of continuous linear operators $S(t): H \rightarrow H$ such that $S(0)=I$, where $I$ is the ...
5
votes
1answer
630 views

Schrödinger Kernels on manifolds

Let $M$ be a compact Riemannian manifold and $\Delta$ be the Laplace-Beltrami operator. It is well-known that the solution operator to the heat equation $e^{t \Delta}$ is smoothing for $t>0$ and ...
5
votes
2answers
2k views

How to find the infinitesimal generator of this semigroup?

Definition 1: Let $X$ be a Banach space. A semigroup is a family $\{T(t)\}_{t\geq 0}$ of continuous linear operators $T(t):X\to X$ such that $(i)\;\;T(0)=I$, where $I$ is the identity operator; $(...
5
votes
2answers
255 views

Version of Hille-Yosida Theorem for non contractive semigroups

We say that a semigroup $\{T(t)\}_{t\geq 0}$ of bounded linear operators on a Banach space $X$ is of type $(M,\omega)$ if there are constants $\omega\geq0$ and $M\geq 1$ such that $$\|T(t)\|_{\mathcal{...
5
votes
1answer
328 views

Proof involving strongly continuous semigroups.

Let $ (T(t))_{t \geq 0} $ be a $ C_{0} $-semigroup on a Hilbert space $ X $ with an infinitesimal generator $ A $, and let $ \rho \in (0,1) $. I want to prove that $ \displaystyle \sup_{t \geq 0} \| ...
5
votes
2answers
2k views

Properties of resolvent operators

I am asked to prove the identities of $(12)$ and $(13)$, which are given on page 438 of the textbook PDE Evans, 2nd edition as follows: THEOREM 3 (Properties of resolvent operators). (i) If $\...
5
votes
1answer
682 views

Exponential of the Laplacian operator as diffusion equation

Let $u$ be a function on a domain $\Omega$ with some fixed boundary condition. I have recently seen a notation $e^{\tau \Delta}u$ as meaning the the time evolution of $u$ by diffusion for a time $\...
5
votes
1answer
104 views

Justifying an equality involving a closed operator $A$

Justify the equality $$A \int_0^\infty e^{-\lambda t} S(t) u \, dt = \int_0^\infty e^{-\lambda t} AS(t) u \, dt$$ used in (16) of §7.4.1. (Hint: Approximate the integral by a Riemann sum and recall $A$...
5
votes
1answer
92 views

Help understand the proof of the Multivalued opertor version of the Hille-Yosida Theorem in Ethier-Kurtz

I am looking at the proof of the following version of the Hille-Yosida theorem given in Ethier and Kurtz' Markov Process. Some Definitions. Here, $A$ is a (multivalued) linear operator on $L$, i.e. ...
5
votes
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 $(\mathcal D(A),A)$ ...
5
votes
1answer
282 views

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
4
votes
2answers
221 views

Generating a $C_0$-semigroup on $L^2$

Consider the linear operator $$A : H^4(\mathbb{R}; \mathbb{R}) \to L^2(\mathbb{R};\mathbb{R})$$ defined by $u\mapsto -(1-\partial_{xx}^2)^2$. Show that $A$ generates a $C_0$-semigroup on $L^2$. I ...
4
votes
1answer
320 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
4
votes
1answer
462 views

If $\{T(t)\}_{t\geq 0}$ is an uniformly continuous semigroup of bounded linear operators then $T(s)\to T(t)$.

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on $X$ ...
4
votes
2answers
146 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 ...
4
votes
2answers
214 views

Show that the fractional power of a linear operator is closed

Let $H$ be a $\mathbb R$-Hilbert space and $(\mathcal D(A),A)$ be a linear operator. Assume $(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ is an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;\;\;\...
4
votes
1answer
346 views

dissipative operator meaning

Can someone explain to me the "meaning" of the dissipative operator ? https://en.wikipedia.org/wiki/Dissipative_operator I am a bit confused. Thanks in advance.
4
votes
1answer
326 views

Prove that $\{S(t)\}_{t \ge 0}$ is not a contraction semigroup on $L^\infty(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the heat ...
4
votes
1answer
742 views

Proof that $\Delta$ generates analytic semigroup

First off, I apologize for asking a question which I'm sure has been studied to death, but I can't seem to find an answer with google. I want to see a proof that the Laplace operator $\Delta$ with ...
4
votes
1answer
827 views

Uniqueness mild solution of $\dot{x} = A x$

Let $A$ be the infinitesimal generator of a $C_0$-semigroup $(S(t))_{t \geq 0}$. Now, for every $x_0 \in X$ the map $t \mapsto S(t) x_0$ is a mild solution of $$ \dot{x} = Ax, \quad x(0) = x_0.\tag{*}...
4
votes
0answers
71 views

Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalents norms?

Do we have that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms This results is pretty easy and straightforward for $p=2$ using ...
4
votes
2answers
92 views

Equivalent Norms for Intermediate Subspaces

Let $(X,\left\|\cdot\right\|)$ be a Banach space, and let $\left\{T(t) : t\geq 0\right\}$ be an equibounded strongly continuous semi-group on $X$. Define a functional $\left\|\cdot\right\|_{\alpha,r;q}...
3
votes
1answer
416 views

Evolution semigroups for differential equations

I would like to ask whether "evolution semigroups" are really useful (to discover something that can't be discovered in some other way?). There is a huge machinery to deal with them, but from my point ...
3
votes
1answer
824 views

What do the operator semigroups have to do with PDE's?

Can anybody please help me to understand what does the semi-group do with partial differential equations? We started this subject very recently and we are now in the proof of Hille-Yosida Theorem, ...
3
votes
2answers
502 views

A spectrum of an invertible operator does not contain $0$

Let $A:D(A)\to X$ be closed linear operator in a Banach space. For some complex number $\lambda$, the resolvent operator $R(\lambda,A)$ is the inverse of the operator $\lambda I-A$ if it exists. In ...
3
votes
1answer
696 views

If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute

Let $A,B$ be complex valued square matrices. If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute. The converse of this statement can be an easy application of the Cauchy ...
3
votes
1answer
125 views

A simple $C_{0}$-semigroup question.

Let $u:[0,t_{e}]\to\mathcal{D}(A)$ satisfy $$\begin{cases} \frac{du}{dt}=Au & 0\le t \le t_{e} \\ u(0)=x \end{cases}$$ I want to prove that necessarily $u(t)=T(t)x$. So it's clear to see that $...
3
votes
2answers
104 views

Understanding the definition of a Integral

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on $X$ ...
3
votes
1answer
38 views

Why the operator $T$ is positive and self-adjoint, which $(T(t)f)=\sum_{n=0}^{\infty}(n+1)^{-t}c_{n}z^n$?

Let $\mathscr{H}=\{f:\mathbb{D}\to \mathbb{C}| f(z)=\sum_{n=0}^{\infty}c_{n}z^n, \Vert f\Vert =\sum_{n}\vert c_{n}\vert^2 < \infty\}$. We define $$(T(t)f)=\sum_{n=0}^{\infty}(n+1)^{-t}c_{n}z^n.$$ ...
3
votes
1answer
143 views

The lack of uniform continuity of shift operator on $L^2$

When studying $c_0-$semigroups, I came accross a statement that if we define shift operator $(S(t)f)(x) = f(x+t)$ for $t>0$ on $L^2(\mathbb{R})$, then $S(t)$ forms a $c_0-$semigroup (that's easy) ...
3
votes
1answer
251 views

what is a symmetric semigroup?

First, I think I know what is a symmetric group roughly from algebra. The group of permutation on a set with $n$ element is denoted by $S_n$, and called the symmetric group on $n$ elements (or $n$ ...
3
votes
1answer
49 views

sub-Markovianity and extensions of $L^2$ semigroup contractions to $L^p$

I just read that a contraction semigroup $(T_t)_{t\geq 0}$ on $L^2(X,m;\mathbb{R})$ space can be extended to a contraction semigroup on $L^p$ for any $p\geq 2$ provided that it satisfies the sub-...
3
votes
1answer
722 views

The proof that every bounded linear operator generates an unique uniformly continuous semigroup.

Let $X$ be a Banach space and $A: X \to X$ a bounded linear operator. So, $A$ is the infinitesimal generator of an uniformly continuous semigroup $\{T(t)\}_{t\geq 0}$ on $X$. The proof, as presented ...
3
votes
1answer
507 views

Proof of a property of the uniformly continuous semigroups.

Let $X$ be a Banach space and $T:=\{T(t)\}_{t\geq 0}$ an uniformly continuous semigroup of bounded linear operators on $X$. So, we know that (i) $T(t)$ is linear and bounded for all $t\geq 0$; (...
3
votes
1answer
51 views

Fundamental theorem of calculus for semigroups

I have a Feller semigroup $(P_t)_{t\geq 0}$. Based on this semigroup I define the linear operator $L = \int_0^tP_s\,ds$ as follows. $$x \mapsto Lu(x) = \int_{0}^t\int u(y) p_s(x,dy)\,ds$$ where $p_s$ ...
3
votes
1answer
40 views

Nonautonomous wave equation of memory type

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation $$u'' - \Delta u + \int\limits_0^t {g(t-s)} \Delta u(s)ds = 0$$ This problem can be written ...
3
votes
1answer
152 views

If $\|(z-A_n)^{-1}-(z-A)^{-1}\|\to 0$ and $e^{-tA}$ decays, will $e^{-tA_n}$ decay for large $n$?

Let $A,A_n$ be closed operators on a Hilbert space $\mathcal H$ and assume that $-(A-{\rm Id}_{\cal H})$ generates a strongly continuous contraction semigroup $-A_n$ generates a strongly continuous ...
3
votes
1answer
33 views

Simple Semigroup (of operator) inequality

I have been able to show that the norm $\| x \|_{1} : = \sup_{t \geq 0} \| T(t)x \|$ is equivalent to the norm $\| x \|$, where $T$ is a bounded $C_0$-semigroup. I now want to show that $T$ is a ...
3
votes
1answer
50 views

Determine generator of $C_0$-semigroup

I try to solve the following problem: Let $X$ locally compact and $a \in C(X)$ such that $\text{Re } a \leq w$. Further let $T(t)f := e^{ta}f$ for all $f \in C_0(X)$ and $t \geq 0$. Determine the ...
3
votes
0answers
65 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 $A^1$ $:...