0
votes
2answers
42 views

Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
3
votes
2answers
67 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 ...
0
votes
0answers
16 views

partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
1
vote
0answers
35 views

Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
1
vote
0answers
95 views

Partial differential equations and semigroups: explanation of an example.

The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem $$\left\{\begin{align*} ...
9
votes
1answer
247 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 ...
1
vote
0answers
29 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
3
votes
0answers
60 views

A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
1
vote
1answer
108 views

Necessary and sufficient condition of being dissipative

I want to know a necessary and sufficient condition on $m:\Omega \mapsto\mathbb{C}$ such that the multiplication operator $M_{m}$ is dissipative in $L^{p}(\Omega)$, where $\Omega$ is a Banach space. ...
1
vote
3answers
204 views

Is a contraction semigroup infinitesimal operator bounded?

Let $T_t:L\to L$ be a semigroup of linear operators $T_t$ acting on a Banach space $L$. Assume that $$ \|T_t\| := \sup\limits_{f\in L}\frac{\|T_tf\|_L}{\|f\|_L} \leq 1 $$ for all $t\geq 0$. The ...