Abstract and Applied Analysis

Volume 2014 (2014), Article ID 327852, 2 pages

http://dx.doi.org/10.1155/2014/327852

Erratum

Erratum to “Compact Operators for Almost Conservative and Strongly Conservative Matrices”

S. A. Mohiuddine,¹ M. Mursaleen,² and A. Alotaibi¹

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 18 April 2014; Accepted 16 June 2014; Published 3 July 2014

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].

References

S. A. Mohiuddine, M. Mursaleen, and A. Alotaibi, “Compact operators for almost conservative and strongly conservative matrices,” Abstract and Applied Analysis, vol. 2014, Article ID 567317, 6 pages, 2014.

M. Mursaleen, “On -invariant mean and -almost convergence,” Analysis Mathematica, vol. 37, no. 3, pp. 173–180, 2011.

M. Mursaleen and A. K. Noman, “On σ-conservative matrices and compact operators on the space Vσ,” Applied Mathematics Letters: An International Journal of Rapid Publication, vol. 24, no. 9, pp. 1554–1560, 2011.