AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/327852 327852 Erratum Erratum to “Compact Operators for Almost Conservative and Strongly Conservative Matrices” Mohiuddine S. A. 1 Mursaleen M. 2 Alotaibi A. 1 1 Department of Mathematics, Faculty of Science King Abdulaziz University P.O. Box 80203, Jeddah 21589 Saudi Arabia kau.edu.sa 2 Department of Mathematics Aligarh Muslim University, Aligarh 202002 India amu.ac.in 2014 372014 2014 18 04 2014 16 06 2014 3 7 2014 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 f and state the results of  in this light.

Let B be a semigroup of positive regular matrices B = ( b n k ) .

A bounded sequence x = ( x k ) is said to be B -almost convergent to the value l if and only if t p n ( x ) l , as p uniformly in n , where (1) t p n ( x ) = 1 p + 1 m = 0 p B m + n ( x ) ; ( p , n N ) , and B n ( x ) = k = 1 b n k x k which is B -transform of a sequence x (see Mursaleen ). The number l is called the generalized limit of x , and we write l = f - lim x . We write (2) f = { x l : lim p t p n ( x ) = L    uniformly in    n } .

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

An infinite matrix A = ( a n k ) n , k = 1 is said to be B -almost conservative if A x f for all x c , and we denote it by A ( c , f ) . An infinite matrix A = ( a n k ) n , k = 1 is said to be B -strongly conservative if A x c for all x f , and we denote it by A ( f , c ) .

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

Theorem 11.

Let A = ( a n k ) be a B -almost conservative matrix. Then, one has (3) 0 L A χ lim sup n ( k = 1 | a ~ n k | ) , L A    i s    c o m p a c t    i f lim n ( k = 1 | a ~ n k | ) = 0 , where a ~ n k = j = 1 a n j b j k .

Proof.

It follows on the same lines as of Theorem 11  by only replacing a n k by a ~ n k .

Theorem 15.

Let B be a normal positive regular matrix. Let A = ( a n k ) be an infinite matrix. Then, one has the following.

If A ( f , c 0 ) , then (4) L A χ = lim sup n ( k = 1 | a ^ n k | ) .

If A ( f , c ) , then (5) 1 2 · lim sup n ( k = 1 | a ^ n k - α k | ) L A χ lim sup n ( k = 1 | a ^ n k - α k | ) ,

where α k = lim n a ^ n k for all k N .

If A ( f , l ) , then (6) 0 L A χ lim sup n ( k = 1 | a ^ n k | ) ,

where A ^ = ( a ^ n k ) is the composition of the matrices A and B - 1 ; that is, a ^ n k = j = 1 a n j b j k - 1 .

Proof.

It follows on the same lines as Theorem 15 of   by only replacing a n k by a ^ n k .

Remark 1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

If B consists of the iterates of the operator T defined on l by T x = ( x σ ( n ) ) , where σ is an injection of the set of positive integers into itself having no finite orbits, then B -invariant mean is reduced to the σ -mean and B -almost convergence is reduced to σ -convergence. In this case, our results are reduced to the results of .

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