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

A strict topology on Orlicz spaces

Let ϕ be a Young function, Ω be a locally compact space, and μ be a positive Radon measure on Ω. We consider a strict topology βϕ (in the sense of Sentilles-Taylor) on the Orlicz function space Mϕ(Ω) and investigate various properties of this locally convex topology. We also study the Orlicz space Mϕ(G) of a locally compact group G with a left Haar measure under the strict topology βϕ and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on Mϕ(G) under the βϕ topology and other natural one