MATRIX TRANSFORMATIONS BETWEEN THE SPACES OF CESÀRO SEQUENCES AND INVARIANT MEANS

Let ω be the space of all sequences, real or complex, and let l∞ and c, respectively, be the Banach spaces of bounded and convergent sequences x = (xn) with norm ‖x‖ = supk≥0 |xk|. Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ on l∞ is said to be an invariant mean or a σmean if and only if (i) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for each n; (ii) φ(e) = 1, where e = (1,1,1, . . .); and (iii) φ((xσ(n)))= φ(x), x ∈ l∞. For certain kinds of mappings, every σmean extends the limit functional φ on c in the sense that φ(x) = limx for x ∈ c (see [2, 15]). Consequently, c ⊂ cσ , where cσ is the set of bounded sequences, all of whose invariant means are equal (see [1, 9, 10]). When σ is translation, the σmeans are classical Banach limits on l∞ (see [2]) and cσ is the set of almost convergent sequences ĉ (see [7]). Almost convergence for double sequences was introduced and studied by Móricz and Rhoades [8] and further by Mursaleen and Savaş [13], Mursaleen and Edely [12], and Mursaleen [11]. If x = (xn), write Tx = (Txn)= (xσ(n)), then


Introduction
Let ω be the space of all sequences, real or complex, and let l ∞ and c, respectively, be the Banach spaces of bounded and convergent sequences x = (x n ) with norm x = sup k≥0 |x k |.Let σ be a mapping of the set of positive integers into itself.A continuous linear functional φ on l ∞ is said to be an invariant mean or a σmean if and only if (i) φ(x) ≥ 0, when the sequence x = (x n ) has x n ≥ 0 for each n; (ii) φ(e) = 1, where e = (1,1,1,...); and (iii) φ((x σ(n) )) = φ(x), x ∈ l ∞ .
For certain kinds of mappings, every σmean extends the limit functional φ on c in the sense that φ(x) = limx for x ∈ c (see [2,15]).Consequently, c ⊂ c σ , where c σ is the set of bounded sequences, all of whose invariant means are equal (see [1,9,10]).When σ is translation, the σmeans are classical Banach limits on l ∞ (see [2]) and c σ is the set of almost convergent sequences c (see [7]).Almost convergence for double sequences was introduced and studied by M óricz and Rhoades [8] and further by Mursaleen and Savas [13], Mursaleen and Edely [12], and Mursaleen [11]. where We define l σ ∞ the space of σbounded sequences (Ahmad et al. [2]) in the following way. Let where ∞ is the set of almost bounded sequences l ∞ (see [14]).Let A = (a nk ) be an infinite matrix of complex numbers a nk (n,k = 1,2,...) and X, Y two subsets of ω.We say that the matrix A defines a matrix transformation from X into Y if for every sequence x = (x k ) ∈ X the sequence A(x) = (A n (x)) ∈ Y , where A n (x) = k a nk x k converges for each n.We denote the class of matrix transformations from X into Y by (X,Y ).
The main purpose of this paper is to characterize the classes (ces[(p),(q)],c σ ) and (ces[(p),(q)],l σ ∞ ) and deduce some known and unknown interesting results as corollaries.The classes (ces[(p),(q)],c σ ) and (ces[(p),(q)],l σ ∞ ) are due to Khan and Rahman [4].If {q n } is a sequence of positive real numbers, then for p = (p r ) with inf p r > 0, we define the space ces[(p),(q)] by where Remark 1.1.If q n = 1 for all n, then ces[(p),(q)] reduces to ces(p) studied by Lim [6].Also, if p n = p for all n and q n = 1 for all n, then ces[(p),(q)] reduces to ces p studied by Lim [5].
For any bounded sequence p, the space ces[(p),(q)] is a paranormed space with the paranorm given by (see [4])

Sequence-to-sequence transformations
In this section, we characterize the classes (ces[(p),(q)],c σ ) and (ces[(p),(q)],l σ ∞ ).We write a(n,k) to denote the elements a nk of the matrix A, and for all integers n,m ≥ 1, we write where t(n,k,m) = 1/(m + 1) m j=0 a(σ j (n),k).We also define the spaces of σconvergent series and σbounded series, respectively, as follows: s and b σ s reduce to c s and b s , as defined below: (2.3) Now we prove the following theorem.
Sufficiency.Suppose that the conditions hold.Fix n ∈ N.For every integer s ≥ 1, from (i) we have (2.6) Now letting s → ∞, we obtain ( Therefore, from (ii) we have Hence (u k ) k and {t(n, k,m)} k ∈ ces * [(p),(q)], therefore the series ∞ k=1 t(n,k,m)x k and ∞ k=1 u k x k converge for each m and n and x ∈ ces[(p),(q)].For given > 0 and x ∈ ces[(p),(q)], choose s such that (2.9) Since (ii) holds, there exists m 0 such that This completes the proof.
Remark 2.1.For different choices of p, q, and σ, we can deduce many corollaries from the above theorem to characterize the matrix classes, for example, (ces(p),c σ ), (ces p ,c σ ), (ces p (q),c σ ), (ces[(p),(q)], c), and so forth.The class (ces(p), c) was characterized by F. M. Khan and M. A. Khan [3] which we can obtain directly from our theorem by taking q n = 1 for all n and σ(n We write (see [2]) where (2.14) Now we prove the following theorem.
This completes the proof.
Remark 2.2.The matrix class (ces(p), l ∞ ), was characterized by F. M. Khan and M. A. Khan [3] which we can obtain directly from the above theorem by letting q n = 1 for all n and σ(n) = n + 1. Besides, we can further deduce many corollaries for different choices of p, q, and σ.