Some Matrix Transformations of Convex and Paranormed Sequence Spaces into the Spaces of Invariant Means

We determine the necessary and sufficient conditions to characterize the matrices which transform convex sequences and Maddox sequences into 𝑉𝜎(𝜃) and 𝑉∞𝜎(𝜃).


Introduction and Preliminaries
By w, we denote the space of all real-valued sequences x x k ∞ k 1 .Any vector subspace of w is called a sequence space.We write that ∞ , c, and c 0 denote the sets of all bounded, convergent, and null sequences, respectively, and note that c ⊂ ∞ ; also cs and p are the set of all convergent and p-absolutely convergent series, respectively, where p : {x ∈ w : In the theory of sequence spaces, a beautiful application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Banach limit.That is, the lim functional defined on c can be extended to the whole ∞ , and this extended functional is known as the Banach limit 1 .In 1948, Lorentz 2 used this notion of a weak limit to define a new type of convergence, known as the almost convergence.Later on, Raimi 3 gave a slight generalization of almost convergence and named it the σ-convergence.
A sequence space X with a linear topology is called a K-space if each of the maps p i : X → C defined by p i x x i is continuous for all i ∈ N. A K-space X is called an FK-space if X is a complete linear metric space.An FK-space whose topology is normable is called a BK-space.An FK-space X is said to have AK property if X ⊃ φ and e k is a basis for X, where e k is a sequence whose only nonzero term is a 1 in kth place for each k ∈ N and φ span{e k }, the set of all finitely nonzero sequences.If φ is dense in X, then X is called an AD-space, thus AK implies AD.For example, the spaces c 0 , cs, and p are AK-spaces.
Let X and Y be two sequence spaces, and let 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 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, . . ., and iii ϕ x ϕ x σ k for all x ∈ ∞ .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. 4 .Consequently, c ⊂ V σ , the set of bounded sequences all of whose σ-means are equal.We say that a sequence x x k is σ-convergent if and only if x ∈ V σ .Using this concept, Schaefer 5 defined and characterized σ-conservative, σ-regular, and σ-coercive matrices.If σ is translation, then V σ is reduced to the set f of almost convergent sequences 2 .
The notion of σ-convergence for double sequences has been introduced in 6 and further studied in 7-9 .
Recently, the sequence spaces V σ θ and V ∞ σ θ have been introduced in 10 which are related to the concept of σ-mean and the lacunary sequence θ k r .In this section, we establish the necessary and sufficient conditions on the matrix A a nk ∞ n,k 1 which transforms r-convex sequences in to the spaces V σ θ and V ∞ σ θ .By a lacunary sequence, we mean an increasing integer sequence θ k r 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 .A bounded sequence x x k of real numbers is said to be σ-lacunary convergent to a number L if and only if lim r → ∞ 1/h r j∈I r x σ j n L, uniformly in n, and let V σ θ denote the set of all such sequences, that is, In this case, L is called the σ, θ -limit of x.We remark that A bounded sequence x x k of real numbers is said to be σ-lacunary bounded if and only if sup r,n | 1/h r j∈I r x σ j n | < ∞, and let V ∞ σ θ denote the set of all such sequences, that is,

Convex Sequence Spaces
Pati and Sinha 12 defined r-convex sequences as follows: The space of all bounded r-convex sequences with r ≥ 2 is denoted by SC r , that is,

2.2
It is clear that SC 1 ⊆ c.
It is well known that Zygmund 13 a bounded convex sequence x k is nonincreasing.It is easy to prove the identity Δ r s x k Δ r Δ s x k , r, s ≥ 0, which shows that SC r ⊂ SC r−1 , when r ≥ 2. Properties of bounded r-convex sequences have been investigated by Rath 14 .Note that SC r ⊂ v ⊂ c ⊂ ∞ .Recently, using the generalized difference operator Δ r , C ¸olak and Et 15 , and Et and C ¸olak 16 defined and studied the sequence spaces c 0 Δ r , c Δ r , and ∞ Δ r .In this section, we establish the necessary and sufficient conditions on the matrix A a nk where for our convenience, we use g n, k, m instead of g r n, k, m for r ≥ 2 throughout the paper. Proof.Sufficiency Suppose that the conditions i and 2.6 hold and x x k ∈ SC r ⊂ ∞ .Therefore, Ax is bounded, and we have Taking the supremum over n, m on both sides and using 2.6 , we get Ax ∈ V ∞ σ θ for x ∈ SC r .

Necessity
Let A ∈ SC r , V ∞ σ θ .Condition i follows as in the proof of Theorem 2.1.Write q n x sup m |λ mn Ax |.It is easy to see that q n is a continuous seminorm on SC r , since for x ∈ SC r ⊂ ∞ , q n x ≤ M x , M > 0. 2.8 Suppose that 2.6 is not true, then there exists x ∈ SC r with sup n q n x ∞.By the principle of condensation of singularities cf.19 , the set is of second category in SC r and hence nonempty, that is, there is x ∈ SC r with sup n q n x ∞.But this contradicts the fact that q n is pointwise bounded on SC r .Now, by the Banach-Steinhaus theorem, there is a constant M such that q n x ≤ M x .

2.10
Now, we define a sequence x x k by

2.11
Then, x ∈ SC r .Applying this sequence to 2.10 , we get 2.6 .This completes the proof of the theorem.

Maddox Sequence Spaces
A linear topological space X over the real field R is said to be a paranormed space if there is a subadditive function g : X → R such that g θ 0, g x g −x , and scalar multiplication is continuous, that is, |α n − α| → 0 and g x n − x → 0 imply g α n x n − αx → 0 for all x, x n in X and α, α n in R, where θ is the zero vector in the linear space X.Assume here and after that x x k is a sequence such that x k / 0 for all k ∈ N. Let p p k ∞ k 0 be a bounded sequence of positive real numbers with sup k p k H and M max{1, H}.The sequence spaces were defined and studied by Et and C ¸olak 16 and Pati and Sinha 12 .If p k p k 0, 1, . . .for some constant p > 0, then these sets reduce to c 0 , c, l ∞ , and l p , respectively.Note that c 0 p is a linear metric space paranormed by where M max 1, sup p k .l ∞ p and c p fail to be linear metric spaces because the continuity of scalar multiplication does not hold for them, but these two turn out to be linear metric spaces if and only if inf k p k > 0. l p is linear metric space paranormed by h 1 x k |x k | p k 1/M .All these spaces are complete in their respective topologies; however, these are not normed spaces in general cf.20 .
In this section, we characterize the matrix classes l p , V σ θ and l ∞ p , V σ θ .Let Ax be defined, then, for all r, n, we write where and a n, k denotes the element a nk of the matrix A.
Theorem 3.1.A ∈ p , V σ θ if and only if there exists B > 1 such that for every n, In this case, the σ, θ -limit of Ax is k u k x k .

Proof. Necessity
We consider the case 1 < p k < ∞.Let A ∈ p , V σ θ .Since e k ∈ p , the condition ii holds.Put f rn x τ rn Ax , since τ rn Ax exists for each r and x ∈ l p , therefore {f rn x } r is a sequence of continuous real functionals on l p and further sup r |f rn x | < ∞ on l p .Now condition i follows by arguing with uniform boundedness principle.The case 0 < p k ≤ 1 can be proved similarly.

Sufficiency
Suppose that the conditions i and ii hold and x ∈ p .Now for every m ≥ 1, we have Thus, the series k t n, k, r x k and k u k x k converge for each r and x ∈ p .For a given ε > 0 and x ∈ p , choose k 0 such that where H sup k p k .Since ii holds, therefore there exists r 0 such that for every r > r 0 .Hence, by the condition ii , it follows that In this case, the σ, θ -limit of Ax is k u k x k .

Proof. Necessity
Let A ∈ ∞ p , V σ θ , then A ∈ ∞ , V σ θ , and the conditions ii and iii follow from Theorem 3 of Schaefer 5 .Now on the contrary, suppose that i does not hold, then there exists N > 1 such that M n ∞.Therefore, by Theorem 3 of Schaefer 5 , the matrix 12 that is, there exists x ∈ ∞ such that Bx / ∈ V σ θ .Now, let y N 1/p k x k , then y ∈ ∞ p and Bx Ay / ∈ V σ θ , which contradicts that A ∈ ∞ p , V σ θ .Therefore, i must hold.

Sufficiency
Suppose that the conditions hold and x ∈ ∞ p , then for every n, Therefore Ax is defined.Now arguing as in Theorem 3.This completes the proof of the theorem. 3.15

Proof. Sufficiency
Let 3.15 hold and that x ∈ p using the following inequality see 21 : |ab| ≤ C |a| q C −q |b| p , 3.16 for C > 0 and a, b, are two complex numbers q −1 p −1 1 , we have Taking the supremum over r, n on both sides and using 3.15 , we get Ax ∈ V ∞ σ θ for x ∈ p , that is, A ∈ p , V ∞ σ θ .

Necessity
Let A ∈ p , V ∞ σ θ .Write q n x sup r |τ r Ax |.It is easy to see that for n ≥ 0, q n is a continuous seminorm on p , and q n is pointwise bounded on p .Suppose that 3.15 is not true, then there exists x ∈ p with sup n q n x ∞.By the principle of condensation of singularities 19 , the set x ∈ p : sup n q n x ∞ 3.18 is of second category in p and hence nonempty, that is, there is x ∈ p with sup n q n x ∞.But this contradicts the fact that q n is pointwise bounded on p .Now, by the Banach-Steinhaus theorem, there is constant M such that q n x ≤ Mg x .

3.19
Now, we define a sequence x x k by

3.20
where 0 < δ < 1 and Then it is easy to see that x ∈ p and g x ≤ δ.Applying this sequence to 3.19 , we get the condition 3.15 .This completes the proof of the theorem.

where q − 1 k p − 1 k 1 .
Theorem 2.1.A ∈ SC r , V σ θ if and only if i sup i,p | ∞ k p a ik | < ∞, ii there exists a constant M such that for s, n 1, 2, . .., Proof.In 17 , a characterization of A ∈ SC r , F B was given, where F B , in the sense of 18 , is the bounded domain of a sequence B B i of matrices B i b i rk .Now, by the setting b i rk 1, we get Ax ∈ V σ θ , and the series k t n, k, r x k and k u k x k converge for x ∈ ∞ p .Hence, by the condition iii , we get