Matrix Transforms of Speed-Summable and Speed-Bounded Sequences

Abstract

First we recall the notions of $A^{\lambda}$-boundedness, $A^{\lambda}$-summability and the absolute $A^{\lambda}$-summability of sequences, and the notion of $\lambda$-reversibility of $A$, where $A$ is a matrix with real or complex entries and $\lambda$ is the speed of convergence, \ie; a monotonically increasing positive sequence. Let $B$ be a lower triangular matrix with real or complex entries, and $\mu=(\mu_{n})$ be another speed of convergence. We find necessary and sufficient conditions for a matrix $M$ (with real or complex entries) to map the set of all $A^{\lambda}$-bounded sequences (for a normal matrix $A$) into the set of all absolutely $B^{\mu}$-summable sequences, and the set of all $A^{\lambda}$-summable sequences (for a $\lambda$-reversible matrix $A$) into the set of all absolutely $B^{\mu}$-summable sequences.