Summary: We determine the conditions for some matrix transformations from

$n\left(\phi \right)$, where the sequence space

$n\left(\phi \right)$, which is related to the

${\ell}_{p}$ spaces, was introduced by

*W. L. C. Sargent* [J. Lond. Math. Soc. 35, 161–171 (1960;

Zbl 0090.03703)]. We also obtain estimates for the norms of the bounded linear operators defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.