arXiv subjects

Thomas Vils Pedersen

Publications and source records attributed to Thomas Vils Pedersen.

4 recordsLinked to original sources

arXiv2016

Generalized Hardy-Cesàro operators between weighted spaces

We characterize those non-negative, measurable functions $ψ$ on $[0,1]$ and positive, continuous functions $ω_1$ and $ω_2$ on $\mathbb R^+$ for which the generalized Hardy-Cesàro operator $$(U_ψf)(x)=\int_0^1 f(tx)ψ(t)\,dt$$ defines a bounded operator $U_ψ:L^1(ω_1)\to L^1(ω_2)$. Furthermore, we extend $U_ψ$ to a bounded operator on $M(ω_1)$ with range in $L^1(ω_2)\oplus\mathbb Cδ_0$. Finally, we show that the zero operator is the only weakly compact generalized Hardy-Cesàro operator from $L^1(ω_1)$ to $L^1(ω_2)$.

math.FA↗

arXiv2013

Properties of derivations on some convolution algebras

For all the convolution algebras $L^1[0,1),\ L^1_{\text{loc}}$ and $A(ω)=\bigcap_n L^1(ω_n)$, the derivations are of the form $D_μ f=Xf*μ$ for suitable measures $μ$, where $(Xf)(t)=tf(t)$. We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure $μ$. Moreover, for all these algebras we show that the extension of $D_μ$ to a natural dual space is weak-star continuous.

math.FA↗

arXiv2011

Compactness and weak-star continuity of derivations on weighted convolution algebras

Let $ω$ be a continuous weight on $\mathbb R^+$ and let $L^1(ω)$ be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from $L^1(ω)$ to its dual space $L^{\infty}(1/ω)$ are exactly the maps of the form $$(D_ϕf)(t)=\int_0^{\infty}f(s)\,\frac{s}{t+s}\,ϕ(t+s)\,ds\qquad\text{($t\in\mathbb R^+$ and $f\in L^1(ω)$)}$$ for some $ϕ\in L^{\infty}(1/ω)$. Also, every $D_ϕ$ has a unique extension to a continuous derivation $\bar{D}_ϕ:M(ω)\to L^{\infty}(1/ω)$ from the corresponding measure algebra. We show that a certain condition on $ϕ$ implies that $\bar{D}_ϕ$ is weak-star continuous. The condition holds for instance if $ϕ\in L_0^{\infty}(1/ω)$. We also provide examples of functions $ϕ$ for which $\bar{D}_ϕ$ is not weak-star continuous. Similarly, we show that $D_ϕ$ and $\bar{D}_ϕ$ are compact under certain conditions on $ϕ$. For instance this holds if $ϕ\in C_0(1/ω)$ with $ϕ(0)=0$. Finally, we give various examples of functions $ϕ$ for which $D_ϕ$ and $\bar{D}_ϕ$ are not compact.

math.FA↗

arXiv2009

A class of weighted convolution Fréchet algebras

For an increasing sequence $(ω_n)$ of algebra weights on $\mathbb R^+$ we study various properties of the Fréchet algebra $A(ω)=\bigcap_n L^1(ω_n)$ obtained as the intersection of the weighted Banach algebras $L^1(ω_n)$. We show that every endomorphism of $A(ω)$ is standard, if for all n\in\mathbb N$ there exists $m\in\mathbb N$ such that $ω_m(t)/ω_n(t)\to\infty$ as $t\to\infty$. Moreover, we characterise the continuous derivations on this algebra: If for all $n\in\mathbb N$ there exists $m\in\mathbb N$ such that $t*ω_n(t)/ω_m(t)$ is bounded on $\mathbb R^+$, then the continuous derivations on $A(ω)$ are exactly the linear maps $D$ of the form $D(f)=(Xf)*μ$ for $f\in A(ω)$, where $μ$ is a measure in $B(ω)=\bigcap_n M(ω_n)$ and $(Xf)(t)=tf(t)$ for $t\in\mathbb R^+$ and $f\in A(ω)$. If the condition is not satisfied, we show that $A(ω)$ has no non-zero derivations.

math.FA↗