All Questions
Tagged with operator-theory semigroup-of-operators
221
questions
2
votes
0
answers
37
views
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 ...
- 21
0
votes
0
answers
48
views
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 $H^{-1}$ ...
0
votes
1
answer
69
views
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 ...
- 23
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 ...
- 6,277
2
votes
1
answer
94
views
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 ...
- 6,277
3
votes
1
answer
119
views
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
$$
A\Gamma =...
- 839
2
votes
1
answer
127
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Exponential boundedness of a strongly continuous semigroup $(T_t)_{t>0}$.
Let $T = (T_t)_{t>0} $ be a strongly continuous semigroup (of bounded operators) on a Banach space $E$, i.e, $ \lim_{t \rightarrow z} \|T_tx- T_{z}x \|, \forall z >0, \forall x \in E $. Note ...
- 603
3
votes
2
answers
221
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Every bounded linear operator is an infinitesimal generator
I'm studying the theory of semigroups from Pazy's book. I'm struggling to understand a specific inequality in the proof of a theorem stating that every bounded linear operator is an infinitesimal ...
- 135
5
votes
0
answers
115
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Existence of a weak-star limit of a family of operators in $B(\mathcal{H})$
Thank you in advance for reading this question, and your thoughts.
I am working with a family of operators $(A_{s,\alpha})_{s\geq 0,\alpha>0}$ in the space $B(\mathcal{H})$ (the space of bounded ...
- 75
2
votes
1
answer
129
views
The Spectrum of the derivative operator in a specific Banach space
Consider the Banach space $X=\left\{u\in C^1([0,1]):\, u(0)=0\right\}$ and the subspace $D=\{u\in C^2([0,1]):\, u(0)=u(1)=u'(0)=0\}$, and the operator $A:D\longrightarrow X$ defined by $Au=u'$. I have ...
- 87
3
votes
1
answer
49
views
Proving that $(G_\lambda)_\lambda$ is a resolvent family
Let $\mathcal{E}$ be a bilinear form on dense subset $\mathcal{D}\subset H$ of a Hilbert space $H$. Assume $\mathcal{E}$ is a closed, symmetric and positive definite, i.e., $\mathcal{E}(u,u)\geq0$. ...
- 24.2k
0
votes
0
answers
24
views
Can we find an explicit solution of this Poisson equation?
Let $\kappa$ be a Markov kernel with invariant measure $\pi$ and $$A:=\kappa-\operatorname{id}$$ denote the corresponding (discrete-time) generator of $\kappa$. Let $c>0$ and $$r:=\frac cp$$ where $...
- 13.9k
0
votes
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-\...
- 13.9k
3
votes
1
answer
185
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)^...
- 123
0
votes
2
answers
88
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 $C_k$ ...
