# Tagged Questions

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### A property of groups of operators

Let $X$ be a Banach space. We consider the evolution equation: $$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$ where $A$ is a bounded operator. I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...
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### Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized Kinetic Energy'. On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...
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Let $G(t,x)$ be the heat kernel $$G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.$$ Here is one approximation to $G(t,x)$: $$G_\epsilon(t,x)=e^{-t/\epsilon} ... 1answer 115 views ### A question about uniformly bounded semigroups Let A be an unbounded linear operator of domain D(A) defined on a Banach space X. Suppose that A generates a C_0-semigroup T(t) which is uniformly bounded. I would like to know if there ... 1answer 93 views ### Integral representation of the resolvent of a semigroup Let T(t) be a C_{0}-semigroup with the generator A. Now, does the so called integral representation of the resolvent$$ (\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt $$hold for ... 1answer 134 views ### Is irreducibility sufficient for uniqueness of invariant distribution for a Feller Semigroup? Let (T_t) be a strongly continuous semigroup of positive operators on C(K), where K is a compact space. Assume also that T_t1 =1  for every t\geq 0. (This is also called a Feller Semigroup.) ... 3answers 204 views ### Does the generator of a 1-parameter group of Banach space isometries know which elements are entire? Let X be a complex Banach space. Let (\sigma_t)_{t \in \mathbb{R}} be a 1-parameter group of linear isometries of X which is strongly continuous i.e. t \mapsto \sigma_t(x) is continuous for ... 1answer 197 views ### positive semigroups and convex operator Let \{Z(t)\}_{t\geq 0} be a strongly continuous positive semigroup on a Banach lattice V. (endowed with ordering \leq). \phi:V\rightarrow V is a convex operator. I want to prove that ... 1answer 125 views ### Relation of spectrum Let X denotes a complex C^*-algebra and \{Z(t)\}_{t\geq 0} is a C_0-Semigroup of operators on X. If for x\in X, I have x=x^* (x is selfadjoint), and its spectrum \sigma(x)\subset ... 1answer 120 views ### what is the meaning of the operator  EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}] the name of this operator is one-parameter semigroup for example  EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]f，  fis the function of \overrightarrow{v} what is ... 1answer 181 views ### fraction power of operators in C_0 semigroup Let X be Banach space, and \{Z(t)\}_{t\geq 0}\subseteq B(X) be the C_0-Semigroup of operators defined on X. Moreover, let A be the infinitesimal generator of \{Z(t)\}_{t\geq 0}. A ... 0answers 59 views ### series representation for *un*bounded perturbations of semigroup generators Let A generate an analytic C_0-semigroup on a Banach space X and B be a relatively compact perturbation, i.e., B is compact as an operator from D(A) (with the graph norm) to X. Then ... 1answer 184 views ### Strongly continuous semigroups that cannot be contractions Let X be a Banach space, and (P_t)_{t \ge 0} a strongly continuous semigroup of bounded operators on X. Using the uniform boundedness principle, it's simple to prove that there are constants ... 0answers 60 views ### Is there a strongly stable semigroup which is not uniformly bounded Let X be a Hilbert space and let (T(t))_{t\geq 0} a C_0-semigroup on X, we recall that: 1- (T(t))_{t\geq 0} is said to be uniformly bounded if there exists M\geq 0 such that for all ... 1answer 170 views ### Embedding a semigroup into a divisible semigroup The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ... 2answers 293 views ### On exponential formula Let T(t),t\geq 0, be a C_0-semigroup on a Banach space X. If A is the infinitesimal generator of T(t),t\geq 0, then$$T(t)x=\lim_{n\infty}(I-\frac{t}{n}A)^{-n}x$$for every x \in X, t\geq ... 0answers 240 views ### Continuity of multiplicative character Let G be a discrete group and \beta (G) denote the Stone-Cech compactification of G, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ... 1answer 134 views ### Generator of a C_0-semigroup restricted to a subspace Suppose we have a decreasing filtration of Banach spaces (E_t)\_{t\geq0}, inclusions V_{s+t,t}:E_{s+t}\to E_t and projections P_{t,s+t}:E_t\to E_{s+t} such that P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}. ... 2answers 384 views ### Do these kernel functions satisfy the semi-group property? Dear Friends, Define the kernel functions for a\ge 1,$$ G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in R\;,  where the constant $C_a$ is some normalization ...
Let $X$ a dual Bancah space (there exists a Banach space $Y$ such that $X=Y'$). A weak* semigroup on $X$ is a semigroup $(T_t)_{\geq 0}$ on $X$ such that, for all $x\in X$, we have ...