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].