Bounded linear operators for some new matrix transformations

In this paper, we define (σ, θ)-convergence and characterize (σ, θ)conservative, (σ, θ)-regular, (σ, θ)-coercive matrices and we also determine the associated bounded linear operators for these matrix classes. AMS Subject Classification (2000) : 46A45, 40H05.


Introduction and preliminaries
We shall write w for the set of all complex sequences x = (x k ) ∞ k=0 .Let φ, ∞ , c and c 0 denote the sets of all finite, bounded, convergent and null sequences respectively; and cs be the set of all convergent series.We write we denote the sequences such that e k = 1 for k = 0, 1, ..., and e (n) n = 1 and e (n) k = 0 (k = n).For any sequence x = (x k ) ∞ k=0 , let x [n] = n k=0 x k e (k) be its n-section.
Note that c 0 , c, and ∞ are Banach spaces with the sup-norm x ∞ = sup k |x k |, and p (1 ≤ p < ∞) are Banach spaces with the norm x p = ( |x k | p ) 1/p ; while φ is not a Banach space with respect to any norm.
A sequence (b (n) ) ∞ n=0 in a linear metric space X is called Schauder basis if for every x ∈ X, there is a unique sequence (β n ) ∞ n=0 of scalars such that x = ∞ n=0 β n b (n) .Let X and Y be two sequence spaces and A = (a nk ) ∞ n;k=1 be an infinite matrix of real or complex numbers.We write Ax = (A n (x)), A n (x) = k a nk x k provided that the series on the right converges for each n.If x = (x k ) ∈ X implies that Ax ∈ Y , then we say that A defines a matrix transformation from X into Y and by (X, Y ) we denote the class of such matrices.
Let σ be a one-to-one mapping from the set N of natural numbers into itself.A continuous linear functional ϕ on the space ∞ is said to be an invariant mean or a σ-mean if and only if (i) ϕ(x) ≥ 0 if x ≥ 0 (i.e.x k ≥ 0 for all k), (ii) ϕ(e) = 1, where e = (1, 1, 1, Throughout this paper we consider the mapping σ which has no finite orbits, that is, σ p (k) = k for all integer k ≥ 0 and p ≥ 1, where σ p (k) denotes the pth iterate of σ at k.Note that, a σ-mean extends the limit functional on the space c in the sense that ϕ(x) = lim x for all x ∈ c, (cf [10]).Consequently, c ⊂ V σ , the set of bounded sequences all of whose σ-means are equal.We say that a sequence where L = σ − lim x, where Using the concept of Schaefer [17] defined and characterized the σ-conservative, σ-regular and σ-coercive matrices.If σ is translation then the σ-mean often called Banach Limit [2] and the set V σ reduces to the set f of almost convergent sequence studied by Lorenz [9].By a lacunary sequence we mean an increasing sequence θ = (k r ) of integers such that k 0 = 0 and h r = k r − k r−1 → ∞ as r → ∞.Throughout this paper the intervals determined by θ will be denoted by I r := (k r−1 − k r ], and the ratio k r k r−1 will be abbreviated by q r (see Fredman et al [8]).Recently, Aydin [1] defined the concept of almost lacunary convergent as follow: A bounded sequence x = (x k ) is said be almost lacunary convergent to the number if and only if lim r 1 h r j∈Ir x j+n = , uniformly in n.

(σ, θ) -Lacunary convergent sequences
We define the following: Definition 2.1.A bounded sequence x = (x k ) of real numbers is said to be (σ, θ) -lacunary convergent to a number if and only if lim r 1 hr j∈Ir x σ j (n) = , uniformly in n, and let V σ (θ),denote the set of all such sequences, i.e where Note that for σ(n) = n + 1, σ-lacunary convergence is reduced to almost lacunary convergence.Results similar to that Aydin [1] can easily be proved for the space V σ (θ).Definition 2.2.A bounded sequence x = (x k ) of real numbers is said to be σ -lacunary bounded if and only if sup r,n | 1 hr j∈Ir x σ j (n) | < ∞, and we let V ∞ σ (θ), denot the set of all such sequences Where Definition 2.4.We say that, infinite matrix A = (a nk ) is said to be (σ, θ)-regular if and only if it is V σ (θ)-conservative and (σ, θ)-lim Ax = lim x for all x ∈ c and we denote this by A ∈ (c, V σ (θ)) reg .Definition 2.5.A matrix A = (a nk ) is said to be (σ, θ)-coercive if and only if Ax ∈ V σ (θ) for all x = (x k ) ∈ ∞ and we denote this by A ∈ ( ∞ , V σ (θ)).

(σ, θ)-conservative matrices and bounded linear operators
In the following theorem we characterize (σ, θ)-conservative matrices and find the associated bounded linear operator.
if and only if it satisfies the condition In this case, the (σ, θ)-limit of Ax is where u = (σ, θ)-lim a and Proof.Sufficiency.Let the conditions hold.Let r be any non-negative integer and x = (x k ) ∈ c.For every positive integer n; write Then we have Since τ rn is obviously linear on c, it follows that τ rn ∈ c and τ rn ≤ A .Now, exists for all x ∈ c (cf [5]).Furthermore, τ n ≤ lim inf r τ rn ≤ A for each n and τ n ∈ c .Thus, each x ∈ c has a unique representation By L(x), we denote the right hand side of the above expression which is independent of n.Now, we have to show that lim Then F rn ∈ c , F rn ≤ 2 A for all r, n, lim r F rn (e) = 0 uniformly in n, and lim r F rn (e k ) = 0 uniformly in n for each k.Let K be an arbitrary positive integer.Then Now applying F rn on both sides of the above equality, we have for all r, n.After choosing fixed K large enough, it is easy to see that the absolute value of each term on the right hand side of (3.1.1)can be made uniformly small for all sufficiently large r.Therefore, lim r F rn (x) = 0 uniformly in n; so that Ax ∈ V σ (θ) and the matrix A is (σ, θ)-conservative.
Necessity.Suppose that A is (σ, θ)-conservative.Then Hence (ii) holds.Now, let x = e.Then (σ, θ)-lim This completes the proof of the theorem.Now, we deduce the following.

(σ, θ)-coercive matrices
We use the following lemma in our next theorem.In this case, the (σ, θ)-limit of Ax is Proof.Sufficiency.Let the conditions hold.For any positive integer Letting r → ∞ and using condition (iii), we get Necessity.Let A be (σ, θ)-coercive matrix.This implies that A is (σ, θ)conservative, then we have condition (i) and (ii) from Theorem 3.1.Now we have to show that (iii) holds.
Suppose that for some n, we have Since A is finite, therefore N is also finite.We observe that since +∞ and A is (σ, θ)-coercive, the matrix B = (b nk ), where b nk = a nk − u k , is also (σ, θ)-coercive matrix.By an argument similar to that of Theorem 2.1 in [6], one can find x ∈ ∞ for which Bx / ∈ V σ (θ).This contradiction implies the necessity of (iii).Now, we use Lemma 4.1 to show that this convergence is uniform in n.Let t rk (n) = j∈Ir [a σ j (n),k − u k ] h r and let T (n) be the matrix (t rk (n)).It is easy to see that H(n) ≤ 2 A for every n; and from condition (ii) lim r t rk (n) = 0 for each k, uniformly in n. and the limit exists uniformly in n, since Ax ∈ V σ (θ).Moreover, this limit is zero since Hence lim r k=1 t rk (n) = 0 uniformly in n; i.e. the condition (iii) holds.This completes the proof of the theorem.

that is, lim rτ
rn (e) exists uniformly in n and lim r τ rn (e) = u uniformly in n, the (σ, θ)-limit of a, since a ∈ V σ (θ).Similarly, lim r τ rn e k = u k , the (σ, θ)-limit of a (k) for each k, uniformly in n.Since {e, e 1 , e 2 , • • • } is a fundamental set in c, and sup r |τ r,n (x)| is finite for each x ∈ c, it follows that lim r τ rn (x) = τ n (x),

u
k x k is defined for every x = (x k ) ∈ ∞ .Let x = (x k ) be any arbitrary bounded sequence.For every positive integer r