On Some New Paranormed Euler Sequence Spaces and Euler Core

Jul., 2010, Vol. 26, No. 7, pp. 1207–1222 Published online: June 15, 2010

DOI: 10.1007/s10114-010-8334-x Http://www.ActaMath.com

Acta Mathematica Sinica,

English Series

Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2010

On Some New Paranormed Euler Sequence Spaces and Euler Core

Gaziosmanpa¸sa University Faculty of Arts and Science Department of Mathematics 60250, Tokat, Turkey

E-mail : serkandemiriz@gmail.com Celal C¸ AKAN1)

Department of Mathematics, ˙In¨on¨u University, Faculty of Education 44280, Malatya, Turkey

E-mail : ccakan@inonu.edu.tr

Abstract In this paper, the sequence spaces er0(u, p) and erc(u, p) of non-absolute type which are the

generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces er

0(u, p) and erc(u, p) are linearly isomorphic to spaces c0(p) and c(p), respectively. Furthermore, the α-, β- and γ-duals of the spaces er

0(u, p) and erc(u, p) have been computed and their bases have been

constructed and some topological properties of these spaces have been investigated. Besides this, the class of matrices (er₀(u, p) : μ) has been characterized, where μ is one of the sequence spaces ∞, c and c0 and derives the other characterizations for the special cases ofμ. In the last section, Euler Core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.

Keywords sequence spaces, matrix transformations, core of a sequence MR(2000) Subject Classification 40A05, 46A45, 40C05

1 Introduction

By ω, we shall denote the space of all real-valued sequences. Any vector subspace of ω is called a sequence space. We shall write _∞, c and c0 for the spaces of all bounded, convergent and

null sequences, respectively. Also by bs, cs, 1 and _p, we denote the spaces of all bounded, convergent, absolutely and p-absolutely convergent series, respectively, where 1 < p <∞.

A linear topological space X over the real fieldR 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, i.e., |α_n− α| → 0 and g(x_n− x) → 0 imply g(α_nx_n− αx) → 0 for all α’s in R

and all x’s in X, where θ is the zero vector in the linear space X.

Assume here and after that u = (u_k) is a sequence such that u_k = 0 for all k ∈ N and (q_k), (p_k) are the bounded sequences of strictly positive real numbers with sup p_k = H and

L = max{1, H}, where N = {0, 1, 2, . . .}. Then, the linear spaces _∞(p), c(p), c0(p) and (p)

Received July 8, 2008, Revised December 8, Accepted June 3, 2009 1) Corresponding author

were defined by Maddox [1] (see also Simons [2] and Nakano [3]) as follows: _∞(p) =  x = (x_k)∈ ω : sup k∈N|xk| pk_<∞_, c(p) =  x = (x_k)∈ ω : lim k→∞|xk− l| pk _{= 0 for some l∈ R}_, c0(p) =  x = (x_k)∈ ω : lim k→∞|xk| pk_{= 0}  , (p) =  x = (x_k)∈ ω : k |xk|pk <∞  ,

which are the complete spaces paranormed by

g1(x) = sup k∈N|xk| pk/L ⇐⇒ inf p k> 0 and g2(x) =   k |xk|pk 1/L ,

respectively. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By F and N_k, we shall denote the collection of all finite subsets ofN and the set of all n∈ N such that n ≥ k, respectively.

Let λ, μ be any two sequence spaces and A = (a_nk) be an infinite matrix of real numbers

a_nk, where n, k ∈ N. Then, we say that A defines a matrix mapping from λ into μ, and we denote it by A : λ→ μ, if for every sequence x = (x_k)∈ λ, the sequence Ax = ((Ax)_n), the

A-transform of x, is in μ, where

(Ax)_n= k

a_nkx_k, n∈ N. (1.1)

By (λ : μ), we denote the class of all matrices A such that A : λ→ μ. Thus, A ∈ (λ : μ) if and only if the series on the right-hand side of (1.1) converges for each n∈ N and every x ∈ λ, and we have Ax ={(Ax)_n}_n∈N∈ μ for all x ∈ λ. A sequence x is said to be A-summable to α if Ax converges to α which is called the A-limit of x. If λ and μ are equipped with the limits

λ-lim and μ-lim, respectively, A∈ (λ, μ) and μ-lim_nA_n(x) = λ-lim_kx_k for all x∈ λ, then we say that A regularly maps λ into μ and write A ∈ (λ, μ)reg. Let x = (xk) be a sequence in C, the set of all complex numbers, and Rk be the least convex closed region of complex plane containing x_k, x_k+1, x_k+2, . . . . The Knopp Core (orK-core) of x is defined by the intersection

of all R_k (k = 1, 2, . . .), (see [4, p. 137]). In [5], it is shown that

K-core(x) =

z∈C

B_x(z)

for any bounded sequence x, where B_x(z) =w∈ C : |w − z| ≤ lim sup_k|x_k− z|. Let E be a subset ofN. The natural density δ of E is defined by

δ(E) = lim

n 1

n|{k ≤ n : k ∈ E}|,

where |{k ≤ n : k ∈ E}| denotes the number of elements of E not exceeding n. A sequence

x = (x_k) is said to be statistically convergent to a number l, if δ({k : |x_k− l| ≥ ε}) = 0 for every ε. In this case we write st-lim x = l, see [6]. By st, we denote the space of all statistically convergent sequences.

In [7], the notion of the statistical core (or st-core) of a complex valued sequence has been introduced by Fridy and Orhan, and it is shown for a statistically bounded sequence x that

st-core(x) = z∈C

C_x(z),

where C_x(z) =w∈ C : |w − z| ≤ st-lim sup_k|x_k− z|. The core theorems have been studied by many authors, for instance see [8–13] and the others.

In [14–16], the Euler sequence spaces er₀, er_c and er_p ,er_∞of non-absolute type, which are the matrix domains of Euler mean Erof order r in the sequence spaces c0, c and _p, _∞, respectively, are introduced, some inclusion relations and Schauder basis for the spaces er₀, er_c, er_p are given, and the α-, β- and γ-duals of those spaces are determined. Recently, Polat and Ba¸sar [17] have introduced the spaces er₀(Δ(m)), er_c(Δ(m)) and er_∞(Δ(m)) of sequences whose m-th order difference are null, convergent and bounded and derived some related results which fill up the gap in the existing literature. Quite recently, Kara, ¨Ozt¨urk and Ba¸sarır [18] have been examined some topological and geometric properties of generalized Euler sequence space. The main purpose of this paper is to introduce the sequence spaces er₀(u, p) and er_c(u, p) of non-absolute type which are the set of all sequences whose Er,u-transforms are in the spaces c0(p) and c(p), respectively; where Er,udenotes the matrix Er,u= (er_nk(u)) defined by

er_nk(u) = ⎧ ⎨ ⎩ _n k  (1− r)n−krku_k, 0≤ k ≤ n, 0, k > n,

where 0 < r < 1. Also, we have constructed the basis and computed the α-, β- and γ-duals and investigated some topological properties of the spaces er₀(u, p) and er_c(u, p). Furthermore, we have defined Euler Core (E_r-core) of a sequence and characterized some class of matrices for which E_r-core(Ax)⊆ K-core(x) and E_r-core(Ax)⊆ st-core(x) for all x ∈ _∞.

2 The Sequence Spaceser₀(u, p) and er_c(u, p) of Non-Absolute Type

In this section, we define the sequence spaces er₀(u, p) and er_c(u, p), and prove that er₀(u, p) and

er_c(u, p) are the complete paranormed linear spaces.

For a sequence space λ, the matrix domain λ_A of an infinite matrix A is defined by

λ_A={x = (x_k)∈ ω : Ax ∈ λ}. (2.1)

In [19], Choudhary and Mishra have defined the sequence space (p) which consists of all sequences such that S-transforms are in (p), where S = (s_nk) is defined by

s_nk= ⎧ ⎨ ⎩ 1, 0≤ k ≤ n, 0, k > n.

Ba¸sar and Altay [20] have recently examined the space bs(p) which is formerly defined by Ba¸sar in [21] as the set of all series whose sequences of partial sums are in _∞(p). More recently, Altay and Ba¸sar have studied the sequence spaces rt(p), r_∞t (p) in [22] and rt_c(p), rt₀(p) in [23] which are derived by the Riesz means from the sequence spaces (p), _∞(p), c(p) and c0(p) of Maddox,

With the notation of (2.1), the spaces (p), bs(p), rt(p), r_∞t (p), rt_c(p) and rt₀(p) may be rede-fined by

(p) = [(p)]_S, bs(p) = [_∞(p)]_S, rt(p) = [(p)]_Rt, r_∞t (p) = [_∞(p)]_Rt, rt_c(p) = [c(p)]_Rt, rt₀(p) = [c0(p)]Rt.

Following Choudhary and Mishra [19], Ba¸sar and Altay [20], Altay and Ba¸sar [22, 23], we define the sequence spaces er₀(u, p) and er_c(u, p), as the sets of all sequences such that Er,u-transforms of them are in the spaces c0(p) and c(p), respectively, that is,

er₀(u, p) =  x = (x_k)∈ ω : lim n→∞  n k=0 _n k  (1− r)n−krku_kx_k pn = 0  and er_c(u, p) =  x = (x_k)∈ ω : lim n→∞  n k=0 _n k  (1− r)n−krku_kx_k− l pn = 0 for some l∈ R  .

In the case (u_k) = (p_k) = e = (1, 1, 1, . . .), the sequence spaces er₀(u, p) and er_c(u, p) are, respectively, reduced to the sequence spaces er₀ and er_c which are introduced by Altay and Ba¸sar [14]. With the notation of (2.1), we may redefine the spaces er₀(u, p) and er_c(u, p) as follows:

er₀(u, p) = [c0(p)]_Er,u and er_c(u, p) = [c(p)]_Er,u. (2.2) Define the sequence y = {y_k(r)}, which will be frequently used, as the Er,u-transform of a sequence x = (x_k), i.e. y_k(r) = k  j=0  k j  (1− r)k−jrju_jx_j, k∈ N. (2.3) Now, we may begin with the following theorem which is essential in the text.

Theorem 1 er₀(u, p) and er_c(u, p) are the complete linear metric space paranormed by g, defined

by g(x) = sup k∈N  k j=0  k j  (1− r)k−jrju_jx_j pk/L .

Proof Since the proof is similar for er₀(u, p) and er_c(u, p), we give the proof only for the space

er₀(u, p). The linearity of er₀(u, p) with respect to the co-ordinatewise addition and scalar mul-tiplication follows from the following inequalities which are satisfied for x, z ∈ er₀(u, p) (see Maddox [27, p. 30]) sup k∈N  k j=0  k j  (1− r)k−jrju_j(x_j+ z_j) pk/L ≤ sup k∈N  k j=0  k j  (1− r)k−jrju_jx_j pk/L + sup k∈N  k j=0  k j  (1− r)k−jrju_jz_j pk/L , (2.4)

and for any α∈ R (see [2])

It is clear that g(θ) = 0 and g(x) = g(−x) for all x ∈ er₀(u, p). Again the inequalities (2.4) and (2.5) yield the subadditivity of g and

g(αx)≤ max{1, |α|}g(x).

Let{xn} be any sequence of the points xn∈ er₀(u, p) such that g(xn− x) → 0 and (α_n) also be any sequence of scalars such that α_n→ α. Then, since the inequality

g(xn)≤ g(x) + g(xn− x)

holds by the subadditivity of g, {g(xn)} is bounded and we thus have

g(αnxn− αx) = sup k∈N  k j=0  k j  (1− r)k−jrju_j(αnxn_j − αx_j) pk/L ≤ |αn− α|g(xn) +|α|g(xn− x),

which tends to zero as n→ ∞. This means that the scalar multiplication is continuous. Hence,

g is paranorm on the space er₀(u, p).

It remains to prove the completeness of the space er₀(u, p). Let{xi} be any Cauchy sequence in the space er₀(u, p), where xi ={x₀(i), x(i)₁ , x(i)₂ , . . .}. Then, for a given ε > 0 there exists a

positive integer n0(ε) such that

g(xi− xj) < ε 2

for all i, j > n0(ε). Using the definition of g we obtain for each fixed k∈ N that

|(Er,u_xi₎

k− (Er,uxj)k|pk/L ≤ sup k∈N|(E

r,u_xi₎

k− (Er,uxj)k|pk/L< ε₂ (2.6) for every i, j ≥ n0(ε) which leads to the fact that {(Er,ux0)_k, (Er,ux1)_k, (Er,ux2)_k, . . .} is a

Cauchy sequence of real numbers for every fixed k∈ N. Since R is complete, it converges, say (Er,uxi)_k → (Er,ux)_k as i→ ∞. Using these infinitely many limits (Er,ux)0, (Er,ux)1, . . . , we

define the sequence{(Er,ux)0, (Er,ux)1, . . .}. From (2.6) with j → ∞, we have

|(Er,u_xi₎

k− (Er,ux)_k|pk/L_≤ ε

2 (i≥ n0(ε)) (2.7) for every fixed k∈ N. Since xi ={x(i)_k } ∈ er₀(u, p) for each i∈ N, there exists k0(ε)∈ N such that

|(Er,u_xi₎

k|pk/L <ε

2 (2.8)

for every k ≥ k0(ε) and for each fixed i∈ N. Therefore, taking a fixed i ≥ n0(ε) we obtain

by (2.7) and (2.8) that

|(Er,u_x)

k|pk/L≤ |(Er,ux)k− (Er,uxi)k|pk/L+|(Er,uxi)k|pk/L< ε

for every k≥ k0(ε). This shows that x∈ er0(u, p). Since{xi} was an arbitrary Cauchy sequence,

the space er₀(u, p) is complete and this concludes the proof.  Note that the absolute property does not hold on the spaces er₀(u, p) and er_c(u, p), since there exists at least one sequence in the spaces er₀(u, p) and er_c(u, p) such that g(x) = g(|x|), where|x| = (|x_k|). This says that er₀(u, p) and er_c(u, p) are the sequence spaces of non-absolute type.

Theorem 2 The sequence spaces er₀(u, p) and er_c(u, p) of non-absolute type are linearly

iso-morphic to the spaces c0(p) and c(p), respectively, where 0 < pk ≤ H < ∞.

Proof To avoid repetition of similar statements, we give the proof only for er₀(u, p). We should show the existence of a linear bijection between the spaces er₀(u, p) and c0(p). With the notation

of (2.3), define the transformation T from er₀(u, p) and c0(p) by x→ y = T x. The linearity of

T is trivial. Furthermore, it is obvious that x = θ whenever T x = θ, and hence T is injective.

Let y∈ c0(p) and define the sequence x ={x_k(r)} by

x_k(r) = 1 u_k k  j=0  k j  (r− 1)k−jr−ky_j, k∈ N. Then, we have g(x) = sup k∈N  k j=0  k j  (1− r)k−jrju_jx_j pk/L = sup k∈N|yk| pk/L_{= g1(y) <}∞.

Thus, we have that x∈ er₀(u, p) and consequently T is surjective. Hence, T is a linear bijection and this says that the spaces er₀(u, p) and c0(p) are linearly isomorphic, as was desired. 

3 The Basis for the Spaces er₀(u, p) and er_c(u, p)

In the present section, we give two sequences of the points of spaces er₀(u, p) and er_c(u, p) which form the basis for those spaces.

Let (λ, h) be a paranormed space. Recall that a sequence (b_k) of the elements of λ is called a basis for λ if and only if, for each x∈ λ, there exists a unique sequence (α_k) of scalars such that h  x− n  k=0 α_kb_k  → 0 as n → ∞.

The seriesα_kb_k which has the sum x is then called the expansion of x with respect to (b_n), and written as x =α_kb_k.

Because of the isomorphism T is onto, defined in the proof of Theorem 2, the inverse image of the basis of those spaces c0(p) and c(p) are the basis of the new spaces er0(u, p) and erc(u, p), respectively. Therefore, we have the following:

Theorem 3 Let λ_k(r) = (Er,ux)_k for all k ∈ N and 0 < p_k ≤ H < ∞. Define the sequence b(k)(r) ={b(k)_n (r)}_n∈N of the elements of the space er₀(u, p) by

b(k)_n (r) = ⎧ ⎨ ⎩ 0, 0≤ n < k, 1 u_n _n k  (r− 1)n−kr−n, n≥ k, (3.1) for every fixed k∈ N. Then

(a) The sequence {b(k)(r)}_k∈N is a basis for the space er₀(u, p), and any x∈ er₀(u, p) has a

unique representation of the form

x =

k

(b) The set {b, b(1)(r), b(2)(r), . . .} is a basis for the space er_c(u, p), and any x∈ er_c(u, p) has

a unique representation of the form

x = lb + k [λ_k(r)− l]b(k)(r), (3.3) where b ={_u1 k} ∞ k=0 and l = lim k→∞(E r,u_x) k. (3.4)

Proof It is clear that{b(k)(r)}_k∈N⊂ er₀(u, p), since

Er,ub(k)(r) = e(k)∈ c0(p), k∈ N, (3.5)

for 0 < p_k ≤ H < ∞, where e(k)is the sequence whose only non-zero term is a 1 in k-th place for each k∈ N.

Let x∈ er₀(u, p) be given. For every non-negative integer m, we put

x[m]= m  k=0

λ_k(r)b(k)(r). (3.6) Then, by applying Er,u to (3.6) under the fact (3.5), we obtain

Er,ux[m] = m  k=0 λ_k(r)Er,ub(k)(r) = m  k=0 (Er,ux)_ke(k) and (Er,u(x− x[m]))_i= ⎧ ⎨ ⎩ 0, 0≤ i ≤ m, (Er,ux)_i, i > m,

where i, m∈ N. Given ε > 0, then there is an integer m0such that

sup i≥m|(E

r,u_x)

i|pk/L< ε

2 for all m≥ m0. Hence,

g(x− x[m]) = sup i≥m|(E r,u_x) i|pk/L≤ sup i≥m0 |(Er,u_x) i|pk/L< ε 2 < ε for all m≥ m0 which proves that x∈ er₀(u, p) is represented as in (3.2).

Let us show the uniqueness of the representation for x∈ er₀(u, p) given by (3.2). Suppose, to the contrary, that there exists a representation x =_kμ_k(r)b(k)(r). Since the linear trans-formation T from er₀(u, p) to c0(p), used in Theorem 2, is continuous, we have at this stage

that (Er,ux)_n= k μ_k(r){Er,ub(k)(r)}_n= k μ_k(r)e(k)_n = μ_n(r) for n∈ N,

which contradicts the fact that (Er,ux)_n= λ_n(r) for all n∈ N. Hence, the representation (3.2) of x∈ er₀(u, p) is unique.

(b) Since{b(k)(r)} ⊂ er₀(u, p) and b∈ c0(p), the inclusion{b, b(k)(r)} ⊂ er_c(u, p) is obviously

true. Let us take x∈ er_c(u, p). Then there uniquely exists an l satisfying (3.4). We thus have

z∈ er₀(u, p) whenever we set z = x−lb. Therefore, we deduce by part (a) of the present theorem that the representation of z is unique. This implies that the representation of x given by (3.3)

4 The α-, β- and γ-Duals of the Spaces er₀(u, p) and er_c(u, p)

In this section, we state and prove the theorems determining the α-, β- and γ-duals of the sequence spaces er₀(u, p) and er_c(u, p) of non-absolute type.

We shall firstly give the definition of α-, β- and γ-duals of sequence spaces and after quoting the lemmas which are needed in proving the theorems given in Section 4.

For the sequence spaces λ and μ, define the set S(λ, μ) by

S(λ, μ) ={z = (z_k) : xz = (x_kz_k)∈ μ for all x ∈ λ}. (4.1) With the notation of (4.1), the α-, β- and γ-duals of a sequence space λ, which are respectively denoted by λα, λβ and λγ, are defined by

λα= S(λ, 1), λβ = S(λ, cs) and λγ = S(λ, bs).

Lemma 4 ([24, Theorem 5.1.1]) A∈ (c0(p) : (q)) if and only if

sup K∈F  n   k∈K a_nkM−1/pk qn <∞, ∃ M ∈ N2. (4.2)

Lemma 5 ([24, Theorem 5.1.9]) A∈ (c0(p) : c(q)) if and only if

sup n∈N

 k

|ank|M−1/pk<∞, ∃ M ∈ N2, (4.3)

∃ (αk)⊂ R  lim_n→∞|ank− αk|qn= 0 for all k∈ N, (4.4)

∃ (αk)⊂ R  sup n∈NN

1/qn

k

|ank− αk|M−1/pk<∞, ∃ M ∈ N2 and ∀ n ∈ N1. (4.5)

Lemma 6 ([24, Theorem 5.1.13])) A∈ (c0(p) : _∞(q)) if and only if sup n∈N   k |ank|M−1/pk _qn <∞, ∃ M ∈ N2. (4.6)

Theorem 7 Let K∈ F and K∗={k ∈ N : n ≥ k} ∩ K for K ∈ F. Define the sets E₁r(u, p)

and E₂r(u) as follows :

E₁r(u, p) =  M>1  a = (a_k)∈ ω : sup K∈F  n    k∈K∗ _n k  (r− 1)n−kr−n u_n anM −1/pk < ∞  , E₂r(u) =  a = (a_k)∈ ω : n  n k=0 _n k  (r− 1)n−kr−n u_n an 

 exists for each n ∈ N. Then

[er₀(u, p)]α= E₁r(u, p), [e_cr(u, p)]α= E₁r(u, p)∩ E₂r(u).

Proof We give the proof only for the space er₀(u, p). Let a = (a_n)∈ ω, and define the matrix

Tr= (tr_nk) via the sequence a = (a_n) by

tr_nk= ⎧ ⎪ ⎨ ⎪ ⎩ _n k  (r− 1)n−kr−n u_n an, 0≤ k ≤ n, 0, k > n.

Bearing in mind the relation (2.3) we immediately derive that

a_nx_n= n  k=0 _n k  (r− 1)n−kr−n u_n anyk = (T r_y) n. (4.7)

We therefore observe by (4.7) that ax = (a_nx_n)∈ 1 whenever x∈ er0(u, p) if and only if

Try ∈ 1 whenever y ∈ c0(p). Then, we derive by Lemma 4 with qn = 1 for all n ∈ N that

[er₀(u, p)]α= E₁r(u, p). 

Theorem 8 Define the sets E₃r(u, p), E₄r(u) and E₅r(u) as follows :

Er₃(u, p) =  M>1  a = (a_k)∈ ω : k  n j=k _j k  (r− 1)j−kr−ja_j u_k  M−1/pk _<∞  , Er₄(u) =  a = (a_k)∈ ω : ∞  j=k _j k  (r− 1)j−kr−ja_j u_k 

 exists for each k ∈ N, Er₅(u) =  a = (a_k)∈ ω : lim n→∞ n  k=0 n  j=k _j k  (r− 1)j−kr−ja_j u_k exists  . Then

[er₀(u, p)]β= Er₃(u, p)∩ E₄r(u), [er_c(u, p)]β= [er₀(u, p)]β∩ E₅r(u).

Proof We give the proof again only for the space er₀(u, p). Consider the equation n  k=0 a_kx_k = n  k=0  1 u_k k  j=0  k j  (r− 1)k−jr−ky_j  a_k = n  k=0 _n j=k _j k  (r− 1)j−kr−ja_j u_k  y_k= (Bry)_n. (4.8) where Br= (br_nk) is defined by br_nk= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n  j=k _j k  (r− 1)j−kr−ja_j u_k , 0≤ k ≤ n, 0, k > n. (4.9)

Thus, we deduce from Lemma 5 with q_n = 1 for all n ∈ N and (4.8) that ax = (a_kx_k)∈ cs whenever x = (x_k)∈ er₀(u, p) if and only if Bry ∈ c whenever y = (y_k)∈ c0(p). Therefore we

derive from (4.3) with q_n= 1 for all n∈ N that

[er₀(u, p)]β = E₃r(u, p)∩E₄r(u). 

Theorem 9 Define the set E₆r(u) as follows :

E₆r(u) =  a = (a_k)∈ ω : _n j=k _j k  (r− 1)j−kr−ja_j u_k  ∈ bs  . Then

[er₀(u, p)]γ = E₃r(u, p)∩ E₄r(u), [er_c(u, p)]γ = [er₀(u, p)]γ∩ E₆r(u).

Proof We see from Lemma 6 with (4.8) that ax = (a_kx_k)∈ bs whenever x = (x_k)∈ er₀(u, p) if and only if Bry ∈ _∞ whenever y = (y_k) ∈ c0(p), where Br = (br_nk) is defined by (4.9).

Therefore, we obtain from (4.6) that

Before stating our corollary on the monotonicity of the spaces er₀(u, p) and er_c(u, p), we give the definition and the lemma concerning the perfectness, normality and monotonicity of a sequence space (see [25, p. 48 and p. 52]).

Definition 1 Let λ be a sequence space. Then, λ is called

(i) Perfect if λ = λαα,

(ii) Normal if y∈ λ whenever |y_k| ≤ |x_k|, k ≥ 1, for some x ∈ λ,

(iii) Monotone provided λ contains the canonical preimages of all its step-spaces.

Lemma 10 Let λ be a sequence space. Then, we have

(i) λ is perfect⇒ λ is normal ⇒ λ is monotone, (ii) λ is normal ⇒ λα= λγ,

(iii) λ is monotone ⇒ λα= λβ.

Combining Lemma 10 and Theorem 8, we have

Corollary 11 The spaces er₀(u, p) and er_c(u, p) are not monotone, and so they are neither

normal nor perfect.

5 Certain Matrix Mappings on the Sequence Spaceser₀(u, p) and er_c(u, p)

In this section, we desire to characterize the matrix mappings from the sequence space er₀(u, p) to some of the known sequence spaces. In what follows, for brevity, we write

˜ a_nk= ∞  j=k _j k  (r− 1)j−kr−j u_k anj

for all k, n∈ N. We also use a similar notation with different letters, and assume that any term with negative subscript is equal to zero.

Suppose throughout that the terms of the infinite matrices A = (a_nk) and B = (b_nk) are connected with the relation

b_nk= ˜a_nk. (5.1)

Now, we may give our basic theorem.

Theorem 12 Let μ be any given sequence space. Then, A∈ (er₀(u, p) : μ) if and only if

A_n ∈ [er₀(u, p)]β for all n∈ N, (5.2)

and

B ∈ (c0(p) : μ), (5.3)

where A_n denotes the n-th row of the infinite matrix A = (a_nk).

Proof Let A∈ (er₀(u, p) : μ) and x∈ er₀(u, p). Then, A_n ∈ [er₀(u, p)]β for all n∈ N. Writing

y = Er,ux, we conclude from the fact A_n ∈ [er₀(u, p)]β that

(Ax)_n= (By)_n for all n∈ N. (5.4) Therefore, By∈ μ for all y ∈ c0(p). Hence B∈ (c0(p) : μ).

Conversely, we assume that (5.2) and (5.3) hold. Then, (5.4) again holds. Therefore, Ax∈ μ for all x∈ er₀(u, p) as was desired which completes the proof. 

Corollary 13 A∈ (er₀(u, p) : _∞) if and only if there exists an integer M > 1 such that sup n∈N   k  ∞ j=k _j k  (r− 1)j−kr−j u_k anj  M−1/pk qn <∞ (5.5) {ank}k∈N∈ {er0(u, p)}β, (5.6) where B = (b_nk) is defined as in (5.1).

Corollary 14 A∈ (er₀(u, p) : c) if and only if (5.5)–(5.6) hold, and there is a sequence α_k of the scalars such that

sup n∈N  k  ∞ j=k _j k  (r− 1)j−kr−j u_k anj  M−1/pk_<∞, _(5.7) sup n∈NN 1/qn k  ∞ j=k _j k  (r− 1)j−kr−j u_k anj− αk  M−1/pk_<∞, _(5.8) lim n  ∞ j=k _j k  (r− 1)j−kr−j u_k anj− αk  qn= 0 for all k∈ N. (5.9)

Corollary 15 A∈ (er₀(u, p) : c0) if and only if (5.5), (5.6) and (5.8), (5.9) hold with αk = 0

for all k∈ N.

By the same technique used in Theorem 12, one can prove the following theorem.

Theorem 16 Let μ be any given sequence space. Then, A∈ (er_c(u, p) : μ) if and only if

A_n ∈ [er_c(u, p)]β for all n∈ N, and

B∈ (c(p) : μ),

where A_n denotes the n-th row of the infinite matrix A = (a_nk) and B = (b_nk) is defined as

in (5.1).

When μ = c, we have

Corollary 17 A∈ (er_c(u, p) : c) if and only if (5.7)–(5.9) and

{ank}k∈N∈ {erc(u, p)}β, lim n   k ∞  j=k _j k  (r− 1)j−kr−j u_k anj− α  qn= 0 hold. 6 E_r-Core

In this section we will consider the sequences with complex entries, and by _∞(C) denote the space of all bounded complex valued sequences.

Following Knopp, a core theorem is characterized by a class of matrices for which the core of the transformed sequence is included by the core of the original sequence. For example, Knopp Core Theorem [4, p. 138] states that K-core(Ax) ⊆ K-core(x) for all real valued sequences x whenever A is a positive matrix in the class (c : c)reg.

Here, we will define Euler core (or E_r-core) of a complex-valued sequence and characterize the class of matrices to yield E_r-core(Ax) ⊆ K-core(x) and E_r-core(Ax) ⊆ st-core(x) for all

x∈ _∞(C).

Now, let us write

tr_n(x) = Er(x) = n  k=0 _n k  (1− r)n−krkx_k.

Then, we can define E_r-core of a complex sequence as follows:

Definition 2 Let H_n be the least closed convex hull containing tr_n(x), tr_n+1(x), tr_n+2(x), . . . .

Then, E_r-core of x is the intersection of all H_n, i.e., E_r-core(x) =

∞ n=1

H_n.

Note that, actually, we define E_r-core of x by the K-core of the sequence (tr_n(x)). Hence, we can construct the following theorem which is an analogue ofK-core, [5].

Theorem 18 For any z∈ C, let G_x(z) =  ω∈ C : |ω − z| ≤ lim sup n |t r n(x)− z|  . Then, for any x∈ _∞,

E_r-core(x) = z∈C

G_x(z).

We prove some lemmas which will be useful to the main results of this section. We define the matrix E = (e_nk) by e_nk= n  k=0 _n k  (1− r)n−krka_nk, n, k∈ N.

Lemma 19 A∈ (_∞: er_c) if and only if

E = sup n  k |enk| < ∞, (6.1) lim n enk= αk for each k, (6.2) lim n  k |enk− αk| = 0. (6.3)

Proof Let x∈ _∞and consider the equality n  j=0  n j  (1− r)n−jrj m  k=0 a_jkx_k = m  k=0 _n j=0  n j  (1− r)n−jrja_jk  x_k, m, n∈ N,

which yields as m→ ∞ that n  j=0  n j  (1− r)n−jrj(Ax)_j= (Dx)_n, n∈ N, (6.4) where D = (d_nk) defined by d_nk= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n  j=0  n j  (1− r)n−jrja_jk, 0≤ k ≤ n, 0, k > n.

Therefore, one can easily see that A ∈ (_∞ : er_c) if and only if D ∈ (_∞ : c) (see [19, p. 223])

and this completes the proof. 

Lemma 20 A∈ (c : er_c)reg if and only if the conditions (6.1) and (6.2) of the Lemma 19 hold

with α_k = 0 for all k∈ N and

lim n

 k

e_nk= 1. (6.5)

Proof Since the proof is easy we omit it. 

Lemma 21 A∈ (st ∩ _∞: er_c)reg if and only if A∈ (c : er_c)reg and

lim n

 k∈E

|enk| = 0 (6.6)

for every E ⊂ N with δ(E) = 0.

Proof Because of c⊂ st ∩ _∞, A∈ (c : er_c)reg. Now, for any x ∈ _∞ and a set E ⊂ N with

δ(E) = 0, let us define the sequence z = (z_k) by

z_k = ⎧ ⎨ ⎩ x_k, k∈ E, 0, k /∈ E.

Then, since z ∈ st0, Az ∈ er₀, where er₀ is the space of sequences which the Er-transforms of them in c0. Also, since _

k

e_nkz_k= k∈E

e_nkx_k,

the matrix D = (d_nk) defined by d_nk= e_nk(k∈ E) and d_nk= 0 (k /∈ E) is in the class (_∞: er_c). Hence, the necessity of (6.6) follows from Lemma 19.

Conversely, let x∈ st∩_∞with st-lim x = l. Then, the set E defined by E ={k : |x_k−l| ≥ ε} has density zero and|x_k− l| ≤ ε if k /∈ E. Now, we can write

 k e_nkx_k = k e_nk(x_k− l) + l k e_nk. (6.7) Since   k e_nk(x_k− l) ≤ x k∈E |enk| + ε · E, letting n→ ∞ in (6.7) and using (6.5) with (6.6), we have

lim n

 k

e_nkx_k = l.

This implies that A∈ (st ∩ _∞: er_c)regand the proof is completed. 

Lemma 22 ([26, Cor. 12]) Let A = (a_nk) be a matrix satisfying_k|a_nk| < ∞ and lim_na_nk

= 0. Then, there exists a y∈ _∞ withy ≤ 1 such that

lim sup n  k a_nky_k = lim sup n  k |ank|. Now, we are ready to give the inclusion theorems.

Theorem 23 Let A∈ (c, er_c)reg. Then, Er-core(Ax)⊆ K-core(x) for all x ∈ ∞ if and only if lim n  k |enk| = 1. (6.8)

Proof The matrix E = (e_nk) satisfies the conditions of Lemma 22, so there exists a y ∈ _∞ withy ≤ 1 such that

 ω∈ C : |ω| ≤ lim sup n  k e_nky_k  =  ω∈ C : |ω| ≤ lim sup n  k |enk|  .

On the other hand, since K-core(y)⊆ B1(0), by the hypothesis,

 ω∈ C : |ω| ≤ lim sup n  k |enk|  ⊆ B1(0) ={ω ∈ C : |ω| ≤ 1} , which implies (6.8).

Conversely, let ω∈ E_r-core(Bx). Then, for any given z∈ C, we can write

|ω − z| ≤ lim sup n |t r n(Ax)− z| = lim sup n  z − k e_nkx_k ≤ lim sup n   k e_nk(z− x_k) + limsup n |z|  1 − k e_nk = lim sup n   k e_nk(z− x_k). (6.9) Now, let lim sup_k|x_k− z| = l. Then, for any ε > 0, |x_k− z| ≤ l + ε whenever k ≥ k0. Hence, one can write that

 k e_nk(z− x_k) =  k<k0 e_nk(z− x_k) +  k≥k0 e_nk(z− x_k) ≤ sup k |z − xk|  k<k0 |enk| + (l + ε)  k≥k0 |enk| ≤ sup k |z − xk|  k<k0 |enk| + (l + ε)  k |enk|. (6.10) Therefore, applying lim sup_nin light of the hypothesis and combining (6.9) with (6.10), we have

|ω − z| ≤ lim sup n   k e_nk(z− x_k) ≤ l + ε,

which means that ω∈ K-core(x). This completes the proof. 

Theorem 24 Let A∈ (st ∩ _∞: er_c)reg. Then, Er-core(Ax)⊆ st-core(x) for all x ∈ ∞ if and

only if (6.8) holds.

Proof Since st-core(x)⊆ K-core(x) for any sequence x [13], the necessity of the condition (6.8) follows from Theorem 23.

Conversely, take ω∈ E_r-core(Bx). Then, we can write again (6.9). Now, if st- lim sup|x_k−

see [28]. Now, we can write  k e_nk(z− x_k) =  k∈E e_nk(z− x_k) + k /∈E e_nk(z− x_k) ≤ sup k |z − xk|  k∈E |enk| + (s + ε)  k /∈E |enk| ≤ sup k |z − xk|  k∈E |enk| + (s + ε)  k |enk|.

Thus, applying the operator lim sup_n and using the condition (6.6) with (6.8), we get that lim sup n   k e_nk(z− x_k) ≤ s + ε. (6.11) Finally, combining (6.9) with (6.11), we have

|ω − z| ≤ st- lim sup

k |xk− z|

which means that ω∈ st-core(x), and proof is completed. 

Acknowledgements We wish to express our close thanks to the referees for their valuable suggestions, thus improving the paper considerably.

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