Abstract and Applied Analysis

Copyright © 2014 S. A. Mohiuddine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We redefine the space and state the results of [1] in this light.

Let be a semigroup of positive regular matrices .

A bounded sequence is said to be -almost convergent to the value if and only if , as uniformly in , where and which is -transform of a sequence (see Mursaleen [2]). The number is called the generalized limit of , and we write . We write

Using the idea of -almost convergence, we define the following.

An infinite matrix is said to be -almost conservative if for all , and we denote it by . An infinite matrix is said to be -strongly conservative if for all , and we denote it by .

Now, we restate Theorem 11 and Theorem 15 of [1] as follows, respectively.

Theorem 11. Let be a -almost conservative matrix. Then, one has where .

Proof. It follows on the same lines as of Theorem 11 [1] by only replacing by .

Theorem 15. Let be a normal positive regular matrix. Let be an infinite matrix. Then, one has the following.(i)If , then (ii)If , then  where for all .(iii)If , then  where is the composition of the matrices and ; that is, .

Proof. It follows on the same lines as Theorem 15 of  [1] by only replacing by .

Remark 1 (see [2]). If consists of the iterates of the operator defined on by , where is an injection of the set of positive integers into itself having no finite orbits, then -invariant mean is reduced to the -mean and -almost convergence is reduced to -convergence. In this case, our results are reduced to the results of [3].