Tagged Questions

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Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
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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 ...
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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 ...
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How can I show $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$ where $S(t)f(x)=e^{-t^2-2tx}f(x+t)$..

I need some help with the following: For every $t\in [0, \infty)$ let $S(t):C_0([0, \infty))\rightarrow C_0([0, \infty))$ be the bounded linear operator given by, $$S(t)f(x)=e^{-t^2-2tx}f(x+t)),$$ ...
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How to show that $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$?

For every $t\in[0, \infty)$ consider the bounded linear operator $S(t):C_{ub}(\mathbb R)\rightarrow C_{ub}(\mathbb R)$ given by $$S(t)u(x)=u(x+t),$$ where $C_{ub}(\mathbb R)$ is the set of all bounded ...
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Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
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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 ...
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 ...