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

Topologically transitive sequence of cosine operators on Orlicz spaces

For a Young function phi and a locally compact second countable group G, let L phi (G) denote the Orlicz space onG. In this paper, we present a necessary and sufficient condition for the topological transitivity of a sequence of cosine operators {Cn}n=1 infinity:={12(Tg,wn+Sg,wn)}n=1 infinity, defined on L phi (G). We investigate the conditions for a sequence of cosine operators to be topologically mixing. Further, we go on to prove a similar result for the direct sum of a sequence of cosine operators. Finally, we give an example of topologically transitive sequence of cosine operators

Bishop’s property (β) and weighted conditional type operators in k-quasi class A*n

An operator T is said to be k-quasi class A*n operator if T*ᵏ (|Tⁿ⁺¹|²/ⁿ⁺¹− |T*|² ) Tᵏ ≥ 0, for some positive integers n and k. In this paper, we prove that the k-quasi class A*n operators have Bishop, s property (β). Then, we give a necessary and sufficient condition for T ⊗S to be a k-quasi class A*n operator, whenever T and S are both non-zero operators. Moreover, it is shown that the Riesz idempotent for a non-zero isolated point λ0 of a k-quasi class A*n operator T say Rᵢ, is self-adjoint and ran(Rᵢ) = ker(T −λ₀) = ker(T −λ₀)*. Finally, as an application in the last section, a necessary and sufficient condition is given in such a way that the weighted conditional type operators on L² (Σ), defined by Tw,u(f) := wE(uf), belong to k-quasi- A*n class.Publisher's Versio

Porous sets and lineability of continuous functions on locally compact groups

Let G be a non-compact locally compact group. In this paper we study the size of the set {(f, g) is an element of A x B : f * g is well-defined on G} where A and B are normed spaces of continuous functions on G. We also consider the problem of the spaceability of the set (C-0 (G) boolean AND (C-0(G) * C-0(G))) \ C-00 (G) and (among other results) we show that, for G = R-n, the above set is strongly c-algebrable (and, therefore, algebrable and lineable) with respect to the convolution product