arXiv subjects

Thomas Kalmes

Publications and source records attributed to Thomas Kalmes.

27 records · Page 2Linked to original sources

arXiv2018

Surjectivity of differential operators and linear topological invariants for spaces of zero solutions

We provide a sufficient condition for a linear differential operator with constant coefficients $P(D)$ to be surjective on $C^\infty(X)$ and $\mathscr{D}'(X)$, respectively, where $X\subseteq\mathbb{R}^d$ is open. Moreover, for certain differential operators this sufficient condition is also necessary and thus a characterization of surjectivity for such differential operators on $C^\infty(X)$, resp. on $\mathscr{D}'(X)$, is derived. Additionally, we obtain for certain surjective differential operators $P(D)$ on $C^\infty(X)$, resp. $\mathscr{D}'(X)$, that the spaces of zero solutions $C_P^\infty(X)=\{u\in C^\infty(X);\, P(D)u=0\}$, resp. $\mathscr{D}_P'(X)=\{u\in\mathscr{D}'(X);\,P(D)u=0\}$ possess the linear topological invariant $(Ω)$ introduced by Vogt and Wagner in [27], resp. its generalization $(PΩ)$ introduced by Bonet and Domański in [1].

math.AP↗

arXiv2017

On Banach spaces of vector-valued random variables and their duals motivated by risk measures

We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces and it is shown that these functionals are Lipschitz continuous. The risk functionals cannot be defined on strictly larger spaces of random variables which creates a particular interest for the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.

math.FA↗

arXiv2015

Graph Laplacians do not generate strongly continuous semigroups

We show that for graph Laplacians $Δ_G$ on a connected locally finite simplicial undirected graph $G$ with countable infinite vertex set $V$ none of the operators $α\,\mathrm{Id}+βΔ_G, α,β\in\mathbb{K},β\ne 0$, generate a strongly continuous semigroup on $\mathbb{K}^V$ when the latter is equipped with the product topology.

math.FA↗

arXiv2014

A short remark on the surjectivity of the combinatorial Laplacian on infinite graphs

Applying a well-known theorem due to Eidelheit, we give a short proof of the surjectivity of the combinatorial Laplacian on connected locally finite undirected simplicial graph $G$ with countably infinite vertex set $V$, established by Ceccherini-Silberstein, Coornaert, and Dodziuk. In fact, we show that every linear operator on $\mathbb{K}^V$ which has finite hopping range and satisfies the pointwise maximum principle is surjective.

math.FA↗

arXiv2014

A remark on the frequent hypercyclicity criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation

We generalize a result for the translation $C_0$-semigroup on $L^p(\R_+,μ)$ about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity criterion due to Mangino and Peris to certain weighted composition $C_0$-semigroups. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster-Lasota equation arising in mathematical biology. Weighted composition $C_0$-semigroups on Sobolev spaces are also considered.

math.FA↗

arXiv2014

A simple characterization of chaos for weighted composition $C_0$-semigroups on Lebesgue and Sobolev spaces

We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_ρ(Ω)$ for an open interval $Ω\subseteq\mathbb{R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(Ω)$ of the Sobolev space $W^{1,p}(Ω)$ for a bounded interval $Ω\subset\mathbb{R}$. These characterizations simplify previously obtained characterization of chaos for these classes of $C_0$-semigroups.

math.FA↗

arXiv2013

Strongly continuous semigroups on some Fréchet spaces

We prove that for a strongly continuous semigroup on the Fréchet space of all scalar sequences, its generator is a continuous linear operator and that the semigroup can be represented as exp(tA) where the exponential series converges in a very strong sense. This solves a problem posed by Conejero. Moreover, we improve recent results of Albanese, Bonet, and Ricker about semigroups on strict projective limits of Banach spaces.

math.FA↗

arXiv2011

The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective

For $d\geq 3$ we give an example of a constant coefficient surjective differential operator $P(D):\mathscr{D}'(X)\rightarrow\mathscr{D}'(X)$ over some open subset $X\subset\R^d$ such that $P^+(D):\mathscr{D}'(X\times\R)\rightarrow\mathscr{D}'(X\times\R)$ is not surjective, where $P^+(x_1,...,x_{d+1}):=P(x_1,...,x_d)$. This answers in the negative a problem posed by Bonet and Domański in \cite[Problem 9.1]{Bonet}.

math.FA↗

arXiv2009

Every P-convex subset of $\R^2$ is already strongly P-convex

A classical result of Malgrange says that for a polynomial P and an open subset $Ω$ of $\R^d$ the differential operator $P(D)$ is surjective on $C^\infty(Ω)$ if and only if $Ω$ is P-convex. Hörmander showed that $P(D)$ is surjective as an operator on $\mathscr{D}'(Ω)$ if and only if $Ω$ is strongly P-convex. It is well known that the natural question whether these two notions coincide has to be answered in the negative in general. However, Trèves conjectured that in the case of d=2 P-convexity and strong P-convexity are equivalent. A proof of this conjecture is given in this note.

math.FA↗