Compact and Fredholm Operators on Matrix Domains of Triangles in the Space of Null Sequences

The matrix domain $X_{A}$ of an infinite matrix $A=(a_{nk}){n,k=0}^{\infty}$ of complex numbers in a subset $X$ of the set $\omega$ of all complex sequences is the set of all $x=(x{k}){k=0}^{\infty}\in \omega$ for which the series $A{n}x=\sum_{k=0}^{\infty}a_{nk}x_{k}$ converge for all $n$ and $Ax=(A_{n}x){n=0}^{\infty}\in X$. Also, if $X$ and $Y$ are subsets of $\omega$ then $(X,Y)$ denotes the set of all infinite matrices that map $X$ into $Y$, that is, $A\in(X,Y)$ if and only if $X\subset Y{A}$. Let $c_{0}$ denote the set sequences $x\in\omega$ that converge to zero, and $T=(t_{nk}){n,k=0}^{\infty}$ and $\tilde{T}=(\tilde{t}{nk}){k,k=0}^{\infty}$ be triangles, that is, $t{nk}=\tilde{t}{nk}=0$ for $k>n$ and $t{nn},\tilde{t}{nn}\not=0$ $(n=0,1,\dots)$. We characterise the class $((c{0}){T},(c{0}){\tilde{T}})$. Furthermore we obtain an explicit formula for the Hausdorff measure of noncompactness of operators $L{A}$ given by a matrix $A\in (c_{0}){T},(c{0}){\tilde{T}})$, that is, for which $L{A}(x)=Ax$ for all $x\in (c_{0}){T}$. From this result, we obtain a characterisation the class of compact operators given by matrices in $((c{0}){T},(c{0}){\tilde{T}})$. Finally we give a sufficient condition for an operator given by a matrix to be a Fredholm operator on $(c{0})_{T}$.