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In other words, if $A$ is the infinitesimal generator of $T$, the mild solution of the abstract inhomogeneous Cauchy problem $$\begin{cases}\dot v =A v +f\\v(0)=0\end{cases}$$ needs not to be $W^{1,1}_{loc}(\mathbb{R}_+, X)$. For instance an $f\in C^0(\mathbb{R}_+,X)$ of the form $f(t):=T(t)x$ for some $x=x(\theta)\in X$, gives $v(t)=tT(t)x=tf(t)$ ...

8

Let me comment on what I know in an open set $\Omega$, trying to control $C$. First of all, the heat semigroup can be expressed through a kernel $p$ which is pointwise dominated by the heat kernel in $\mathbb{R}^n$, written by Bazin, but this is not sufficient, because of $\nabla_x$. One can extend the Gaussian bound to complex times thus getting $$|p(z,x,y)... 8 The problem is discussed in a more general setting (operator ideals in Banach spaces) for the so-called analytic semigroups (parabolic problems) in Blunck, S.; Weis, L., Operator theoretic properties of differences of semigroups in terms of their generators, Arch. Math. 79, No. 2, 109-118 (2002). ZBL1006.47036. The paper seems to be freely accessible. The ... 7 This cannot happen, if, e.g., Y is a countable union of finite dimensional subspaces (by the Baire category theorem). Suitable examples are given by the spaces of finite sequences in \ell^p. Edit in response to questioner's comment. If Y is the domain of a closed operator into a Banach space, then the graph topology makes it into a Banach space ... 7 One can prove a stronger estimate in fact. Suppose u and v are positive with v>u say. Note that$$ |G(u,x)-G(v,x)| \le \int_{u}^v \Big| \frac{d}{dt} G(t,x)\Big| dt =\int_{u}^{v} \frac{e^{-x^2/2t}}{\sqrt{2\pi t}} \Big|\frac{x^2}{2t^2}-\frac{1}{2t}\Big| dt. $$Integrating this over x\in {\Bbb R} we obtain$$ \int_{-\infty}^{\infty} |G(u,x)-G(...

7

Following ideas of John von Neumann, J. von Neumann, Über einen Satz von Herrn M. H. Stone, Ann. Math. (2) 33, 567-573 (1932). ZBL0005.16402, it can be shown that if a function satisfies the semigroup law and is measurable, then it is already continuous. Hence your identity is only true if the generator is bounded. See also Engel, Klaus-Jochen; Nagel, ...

7

This question was considered by Functional Analysis specialists around 1960 or earlier. In the case of Banach Algebras $C(X)$ (for Hausdorff compact $X$) this gets reduced often to studying the auto-homeomorphisms of $X$, and it is extra interesting when the compact space is nice. In this context, in 1961, I have rediscovered that orientation preserving ...

7

The purpose of this answer is to extend Christian Remling's answer to dimension $d = 3$. There are two steps. (N.B. below the cut I show how to replace part 1 by a different argument that works in all dimensions, and so this should answer the question posed.) We assume that we have a solution $\phi$ to the Schrodinger equation such that it vanishes on a (...

6

If I am not mistaken, then this is connected to the notion of analytic vectors and related object. If I understand your question correctly, it is answered in this paper of Chernoff. Further, for group generators on Banach spaces the analytic vectors are dense and they determine the generator $D$, see Exercise II.3.12(2) in Engel-Nagel.

6

If I understand your question correctly, this would mean that the function $t\mapsto T(t)x$ is Lipschitz continuous, which is equivalent for $x$ to be in the Favard space $\text{Fav}(A)$$. See for example Defintion 8.2. in these lecture notes. If your space X is reflexive, then the Favard space is exactly D(A), the domain of A, see Corollary II.5.21 ... 6 There is a one-dimensional counterexample: Consider the analytic semigroup z \mapsto e^{iz}. This semigroup is bounded on the non-negative real line, but it is not bounded on any sector \Delta_\delta. EDIT in response to the comments: One can "modify" each analytic semigroup to obtain the following boundedness property: Let 0 < \delta' < \delta ... 6 Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by \Delta+Bx \cdot \nabla, where B is a matrix. Assuming that all eigenvalues of B have negative real parts, then an invariant measure \mu exists (and is given by a Gaussian ... 6 For the record, the result requested in the question is given in Theorem 1.3.3 in: E.B. Davies, Heat Kernels and Spectral Theory, DOI:10.1017/CBO9780511566158. The assumptions are: L is a positive definite self-adjoint operator given by a quadratic form Q on the Hilbert space L^2(\Omega), where \Omega is a \sigma-finite measure space; if u is in ... 6 Probabilistic approach I have to think about the right reference, but things are very much different in the non-local case. The solution u = u^\epsilon can be written in probabilistic terms as$$ u(x) = \mathbb E^x e^{-k \tau_D} ,$$where k = \lambda / \epsilon, \tau_D is the first exit time from the open set D = \Omega (when probability enters, I ... 5 Take K = \{0,1\}^{\mathbf{Z}^2} and take for T_t the Glauber dynamic for the Ising model below the critical temperature. Then T_t is Feller and irreducible, but it has two distinct ergodic invariant measures. If however you know that T_t is strong Feller (or asymptotically strong Feller) then irreducibility does imply uniqueness of the invariant ... 5 An operator A is a group generator if and only if both A and -A are semigroup generators. So under your assumptions, if we denote by \tilde{A} the generator of the extension \{\tilde{T}(t),t\ge 0\} of \{T(t),t\ge 0\} to Y, then -\tilde{A} would be a semigroup generator, too. If by "extend" you mean the usual thing (viz, that X is left ... 5 You remarked in comments that T should be an unbounded operator from X to X. If X is separable then D(T) has to be a Borel set in X. Note that X \times X is a separable Banach space under a norm such as \|(x_1, x_2)\| = \|x_1\| + \|x_2\|. The graph \Gamma(T) = \{(x, Tx) : x \in D(T)\} is by assumption a closed linear subspace of X \... 5 As you write it, this is just a bounded perturbation of the sectorial operator$$\begin{pmatrix} 0 & 0 & 0 \\ 0 & \Delta & 0 \\ 0 & 0 & 0 \end{pmatrix}.$$It is quite standard that bounded perturbation does not destroy this property. 5 First, the classical Calderón–Zygmund estimate gives$$\left\|D^2 u\right\|_{L^p(\mathbb{R}^d)}\leq C(p,d) \left\|\Delta u\right\|_{L^p(\mathbb{R}^d)}.$$By interpolation, \left\|u\right\|_{W^{2,p}(\mathbb{R})} is equivalent to$$\left\|\Delta^2 u\right\|_{L^p(\mathbb{R}^d)}+\left\|u\right\|_{L^p(\mathbb{R}^d)}.$$For detail proof you can look up ... 5 Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see Proposition V.1.7. in Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036. Assuming you ... 5 To complete Piero's answer, let me give a definition of dispersiveness. To stay at a rather simple level, let me consider an linear evolution PDE with constant coefficients,$$P(\partial_t,\nabla_x)u=0.$$Hereabove, P is a polynomial. Let me assume a property of homogeneity,$$P(\lambda^\alpha\tau,\lambda\xi)\equiv\lambda^m P\tau,\xi).$$It is met by the ... 5 The estimate the OP is looking for is called an ultracontractivity estimate. A characterisation of semigroups that satisfy such an estimate can be found in the Theorem on page 65, Subsection 7.3.2, of the following survey article: Wolfgang Arendt: Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates in the Handbook of ... 4 This does not answer your question only adds something to your list of examples: two parameter semigroups, as investigated by Slocinski, M. Unitary dilation of two-parameter semi-groups of contractions. (English) [J] Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 22, 1011-1014 (1974) and many others later on. They are significantly different ... 4 Just to expand the answer by Alex, let me mention that the internet seminar of last year: http://isem17.unisa.it/w/index.php/Phase_1:_The_lectures was intended as a gentle introduction where a lot of time was spent on finite dimensional theory to motivate the ideas in a simpler situation. Though it stresses the positivity aspects of spectral theory, I am ... 4 Apart from the fact that I do not understand some parts of your question, self-adjoint elements with positive spectrum define a positive cone in your$C^\ast$algebra. Positivity-preserving semigroups in$C^\ast$and von Neumann algebras were extensively studies, you should consult the chapter written by Ulrich Groh in the book W. Arendt, A. Grabosch, G. ... 4 This is a symbol which reminds that the solution of linear system of ODEs is represented as an exponential function of a matrix. This symbol represents the solution family of the heat equation, and indeed called a one-parameter semigroup (or semi-dynamical system). Great introduction in the subject are the books of Engel and Nagel, the original one and a ... 4 Toric varieties in Algebraic Geometry!! Indeed, the category of normal toric varieties is equivalent with the dual of the category of finitely generated, integral semigroups. 4 It is well known in the theory of operator semigroups that the Cauchy problem for the generator$D$of a strongly continuous SEMIgroup is always UNIQUELY solvable for initial data in the domain of$D$, that is$[0,\infty)\to X$,$t\mapsto \sigma_t(x)$is the unique solution of$f'(t)=D(f(t))$for all$t\ge 0$and$f(0)=x$. Therefore, if you find by some ... 4 This is standard, but the argument is short enough to fit in an answer. It is not restrictive to assume that$0\in\Omega$. Denote by$Q(t,x,y)$the heat kernel associated with the Dirichlet Laplacian on$\Omega$(this is simply the solution to the heat equation on$\Omega$with Dirichlet b.c. and the delta function as initial data) and by$P(t,x,y)=ct^{-n/2}\...

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