Characterization of Some Classes of Compact Operators Between Certain Matrix Domains of Triangles

Abstract

In this paper, we characterize the classes $((\ell_{1})_{T},(\ell_{1})_{\tilde{T}})$ and $(c_{T},c_{\tilde{T}})$ where $T=(t_{nk})_{n,k=0}^{\infty}$ and $\tilde{T}=(\tilde{t}_{nk})_{n,k=0}^{\infty}$ are arbitrary triangles. We establish identities or estimates for the Hausdorff measure of noncompactness of operators given by matrices in the classes $((\ell_{1})_{T},(\ell_{1})_{\tilde{T}})$ and $(c_{T},c_{\tilde{T}})$. Furthermore we give sufficient conditions for such matrix operators to be Fredholm operators on $(\ell_{1})_{T}$ 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 [I. Djolovi\'{c}, E. Malkowsky, A note on Fredholm operators on $(c_{0})_{T}$, {\it Applied Mathematics Letters }

{\bf 22 }(2009) 1734—1739] and some of them are generalization for those in

[B. de Malafosse, V. Rako\v{c}evi\'c, Application of measure of noncompactness in operators on the spaces $s_{\alpha}$, $s_{\alpha}^{0}$, $s_{\alpha}^{c}$, $\ell_{\alpha}^{p}$ {\it Journal of Mathematical Analysis and Applications} {\bf 323(1)} (2006) 131—145].