Applications of uniform boundedness principle to matrix transformations

Using the uniform boundedness principle of Maddox, we characterize matrix transformations from the space $(\ell_{p}) _{T}$ to the spaces $m(\phi )$ and $n(\phi )$ for the case $1\leq p\leq \infty$, which correspond to bounded linear operators. Here $(\ell _{p})_{T}$ is the domain of an arbitrary triangle matrix $T$ in the space $\ell _{p}$, and the spaces $m(\phi )$ and $n(\phi )$ are introduced by W.L.C. Sargent. In special cases, we get some well known results of W.L.C. Sargent, M. Stieglitz and H. Tietz, E. Malkowsky and E. Savaş. Also we give other applications including some important new classes.