Convex-Cyclic Weighted Translations On Locally Compact Groups

A bounded linear operator $T$ on a Banach space $X$ is called a convex-cyclic operator if there exists a vector $x \in X$ such that the convex hull of $Orb(T, x)$ is dense in $X$. In this paper, for given an aperiodic element $g$ in a locally compact group $G$, we give some sufficient conditions for a weighted translation operator $T_{g,w}: f \mapsto w\cdot f*\delta_g$ on $\mathfrak{L}^{p}(G)$ to be convex-cyclic. A necessary condition is also studied. At the end, to explain the obtained results, some examples are given

On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Let be a locally compact group with a fixed left Haar measure and Ω be a system of weights on . In this paper, we deal with locally convex space (,Ω) equipped with the locally convex topology generated by the family of norms (‖.‖,)∈Ω. We study various algebraic and topological properties of the locally convex space (,Ω). In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which (,Ω) with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum

Porosity of certain subsets of Lebesgue spaces on locally compact groups

Let $G$ be a locally compact group. In this paper, we show that if $G$ is a non-discrete locally compact group, $p\in(0,1)$ and $q\in(0,+\infty]$, then the set of all pairs $(f, g)\in L^p(G)\times L^q(G)$ for which $f\ast g$ is finite, forms a set of first category in $L^p(G)\times L^q(G)$. 10.1017/S000497271200094

Porosity and the lp-conjecture for semigroups

In this paper, we consider the size of the set ( f, g) ∈ p(S) × q (S) : ∃ x ∈ S, | f |∗|g|(x) 0. By means of this notion of porosity we also provide a strengthening of a famous result by Rajagopalan on the p-conjecture