Characterization of some classes of compact operators between certain matrix domains of triangles

Abstract

In this paper, we characterize the classes ((1) T , (1)T) and (c T , cT) where T = (t nk) ∞ n,k=0 and T = (t nk) ∞ n,k=0 are arbitrary triangles. We establish identities or estimates for the Hausdorff measure of noncompactness of operators given by matrices in the classes ((1) T , (1)T) and (c T , cT). Furthermore we give sufficient conditions for such matrix operators to be Fredholm operators on (1) T and c T. As an application of our results, we consider the class (bv, bv) and the corresponding classes of matrix operators. Our results are complementary to those in [2] and some of them are generalization for those in [3].

Some general results on fractional Banach sets

“…Note that well known characterizations of the classical sequence spaces can be found in [29]. Lemma 3.1 [9,Lemma 2.3] Let µ be an arbitrary subset of ω and λ be a BK space with AK . Also let S = (s jk ) be the inverse of an infinite matrix A and R be the transpose of S .…”

“…The sets of sequences ℓ 1 (Ψ) = {σ ∈ ω : Ψσ ∈ ℓ 1 } , c 0 (Ψ) = {σ ∈ ω : Ψσ ∈ c 0 } , and c(Ψ) = {σ ∈ ω : Ψσ ∈ c} were considered in the papers [9,10], where ω denotes the space of all complex valued sequences. Certain results on matrix mappings and compact operators on ℓ 1 (Ψ), c 0 (Ψ), and c(Ψ) were generalized using measures of noncompactness in those papers.…”