# 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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### Commutation of $c_0$-semigroups

Task is: given two $c_0$-semigroups $T(t)$ and $S(t)$ in Banach space X with property $\forall \;t \geq 0 : T(t)S(t) = S(t)T(t)$, show that $\forall \; t, s \geq 0 : T(t)S(s) = S(s)T(t)$. I've tried ...
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### The semi-group property of Riesz potentials in higher dimensions

I'm trying to wrap my head around the Riesz potential in the sense of a higher dimensional generalization of Riemann-Liouville fractional integrals but some things are coming across as somewhat ...
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### What is a $C^\infty$ regularizing contraction semigroup?

could anyone tell me what a $C^\infty$ regularizing contraction semigroup is? I know the contraction semigroup part, but I don't understand what exactly the $C^\infty$ regularizing part. De ...
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### What can we infer from a growth bound What can we infer from the condition $\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|}n\le0$?

Let $A_n$ be a compact linear operator on a $\mathbb R$-Hilbert space $H$. What can we infer from the condition $$\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|_{\mathfrak L(H)}}n\le0\tag1$$ and in ...
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### When are spectral values a pole of the resolvent?

Let $(T_t)_{\geq 0}$ be a $C_0$-semigroup on a Banach space $X$ with the generator $A.$ What are some conditions sufficient for $\lambda \in \sigma(A)$ to be a pole of resolvent? I'm looking for ...
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### When does strong convergence imply convergence in operator norm?

I have a $C_0$-semigroup $(T_t)_{t\ge0}$ and I want to show $\lim_{t \to \infty} T_t =0$ with respect to the operator norm. After some effort, I was able to prove $\lim_{t \to \infty} T_t =0$ ...
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### Range of strong limit of a semigroup belongs to the fixed space?

Let $\left(T(t)\right)_{t\geq0}$ be a $C_0$-semigroup on a Banach lattice $E$ such that $T(t)$ converges strongly to a positive operator $S$ as $t \to \infty.$ Then $$T(t)S=S \text{ for all }t\geq0.$$ ...
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### Characterization of space $D(A^2)$ in semigroup theory for laplacian operator

Let us assume $\Omega$ is sufficient smooth. Let $H=L^2(\Omega)$ and define $A:D(A)\subset H\to H$ by $Au=\Delta u$, with $D(A)=H_0^1(\Omega)\cap H^2(\Omega).$ Brezis's book, Brezis, Functional ...
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### Analytic Semigroups and Regularity

Suppose $L$ is a second order elliptic differential operator with domain $D(L) = W^{2, p}(\mathbb{R}^d)$ generating an analytic semigroup $(T_t)_{t \geq 0}$ in $L^p(\mathbb{R}^d)$ as in "Lundari - ...