A linear subspace of the space of all complex sequences, denoted by , is called a -space if it is a Banach space with continuous coordinates , where is the complex field and for all . Let be an infinite matrix with complex entries and let be the sequence in the th row of for every . If , then the -transform of is the sequence , where , provided that the series on the right converges for each . Let and be subsets of . Then defines a matrix mapping from into if exists and is in for all . Let be the set of all finite complex sequences that terminate in zeros. If and are -spaces, then every infinite matrix that maps into defines a continuous linear operator by for all . Let be a -space, then the matrix domain is also a -space with the norm for all .
In the paper under review, the authors prove some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators that map an arbitrary -space into the -spaces and and into the matrix domains of infinite triangles matrices , i.e., such that the complex entries of satisfy and for all . Further, the authors give necessary and sufficient (or only sufficient) conditions for such operators to be compact.