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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Density of $H^1$ functions with bounded gradient

I have been working on a PDE problem and I came across the need of working with functions with bounded gradient. However, since I am working with semigroups, I need density of the domain of the ...
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Weak limits in continuous convolution semigroups

A convolution semigroup is a collection of probability measures $(\mu_t)_{t \in I}$ on $\mathbb R^d$, where $I \subset [0,\infty)$, for which $\mu_s * \mu_t = \mu_{s+t}$. The convolution semigroup is ...
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The dual semigroup is equivalent in norm to its original semigroup

I would like to show the following inequality regarding the dual semigroup of a semigroup of linear operators (the one at the end of the image). The screenshot comes from the book One-paramter ...
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The heat semigroup represented as $\{e^{t\Delta}; t>0\}$

Set $\{G(t);t>0\}$ the semigroup defined as $G(t):L^2 \to L^2$ for every $t>0$, where $$G(t)u=g_t*u$$ and $g_t(x)=(4\pi t)^{-\frac{n}{2}}\exp(-|x|²/4t)$, $\forall x \in \mathbb{R}^n$. On the ...
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Motivation of the Proof of the Hille-Yosida Theorem

Let $X$ be a Banach space and $A$ be a linear map from a subspace of $X$ to $X$. The Hille-Yosida theorem gives a necessary and sufficient condition for $A$ to be an infinitesimal generator of a ...
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If is $P(S(4^{-k}))$ true so $P(S(t))$ is too true?

i am reading an article about Besov-Morrey spaces an there has the following sentense: "The semigroup propety of $e^{t\Delta}$ allows us to reduce easily to $t = 4^{−k}$ for $k\in \mathbb{Z}$&...
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A step in the proof of the Hille-Yosida theorem from Rudin

I'm getting stuck on perhaps a simple step in the Hille-Yosida theorem from 13.37 in Rudin's functional analysis. I wonder if someone has had this same difficulty before or knows how to get around it -...
Let $A \colon D(A) \subset H \to H$ be the generator of a $C^0$ semigroup. Suppose that $e_k$ is an orthonormal basis and $A$ is diagonalizable $A e_k=\lambda_k e_k$ with eigenvalues $e_k$. Then is it ...