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Representations of functionals on absolute weighted spaces and adjoint operators
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Original article
Representations of functionals on absolute weighted spaces and adjoint operators Mehmet Ali Sarıgöl Department of Mathematics, Pamukkale University, Denizli, Turkey
Received 12 May 2017; received in revised form 8 September 2017; accepted 5 October 2017 Available online xxxx
Abstract In the present paper, we establish general representations of continuous linear functionals, which play important roles in Functional Analysis, of the absolute weighted spaces which have recently been introduced in Sarıgöl (2016, 2011), and also determine their norms. Further making use of this we give adjoint operators of matrix mappings defined on these spaces. c 2017 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V. This is an open access article under the CC ⃝ BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Sequence spaces; BK spaces; Absolute summability; Continuous linear functional
1. Introduction Any vector subspace of w, the space of all (real- or) complex valued sequences, is called a sequence space. A sequence space X is a B K -space if it is a Banach space provided that each of the maps pn : X → C defined by pn (x) = xn is continuous for all n ≥ 0. A B K -space X is said to have AK property if φ ⊂ X and {e(n) } is a basis for X , where e(n) is a sequence whose only non-zero term is 1 in kth place for each n ≥ 0 and φ =span{e(n) }, the set of all finitely non-zero sequences. For example, ℓk , the space of all k-absolutely convergent series, is AK -space for k ≥ 1. Let X, Y be sequence spaces and A = (anv ) be an infinite matrix of complex numbers. If Ax = (An (x)) ∈ Y for every x ∈ ∑ X, then we say that A defines a matrix transformation from X into Y, and denote it by A ∈ (X, Y ), where An (x) = ∞ v=0 anv x v , provided that the series converges for n ≥ 0. Now, let Σ av be a given infinite series with nth partial sums (sn ) and (θn ) be a sequence of nonnegative terms. Then the series Σ av is said to be summable |A, θn |k , k ≥ 1, if ∞ ∑
θnk−1 |∆An (s)|k < ∞, A−1 (s) = 0,
n=0
E-mail address: [email protected]. Peer review under responsibility of Journal Transactions of A. Razmadze Mathematical Institute. https://doi.org/10.1016/j.trmi.2017.10.003 c 2017 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND 2346-8092/⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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M.A. Sarıgöl / Transactions of A. Razmadze Mathematical Institute (
)
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∆An (s) = An (s) − An−1 (s) [1].⏐ If A is the⏐ matrix of⏐ weighted mean N , pn (r esp. θn = Pn / pn ) , then summability ⏐ |A, θn |k reduces to summability ⏐ N , pn , θn ⏐k (r esp. ⏐ N , pn ⏐k , [2]), [3]. Further, if θn = n for n ≥ 1 and A is the matrix of Cesàro mean (C, α), then it is the same as summability |C, α|k in Flett’s notation [4]. By a weighted mean matrix we state { pv /Pn , 0 ≤ v ≤ n anv = (1.1) 0, v>n (
)
where ⏐ ⏐ ( pn ) is a sequence of positive numbers with Pn = p0 +⏐ p1 + · · ·⏐ + pn → ∞ as n → ∞. In [5], the space ⏐ θ⏐ ⏐N p ⏐ , k ≥ 1, was defined as the set of all series summable by ⏐ N , pn , θn ⏐k , i.e., k ⎫ ⎧ ⏐k ⏐ n ∞ ⏐ ⎬ ⏐ ⎨ ⏐ ⏐ ∑ ∑ ⏐ ⏐ ⏐ θ⏐ Pv−1 av ⏐ < ∞ , ⏐γn ⏐N p ⏐ = a = (av ) : ⏐ ⎭ ⏐ ⎩ k n=1
v=1
which is a BK-space with respect to the norm (see [6]) ⎧ ⏐k ⎫ k1 ⏐ n ∞ ⏐ ⎨ ⏐ ⎬ ∑ ∑ ⏐ ⏐ ∥a∥⏐⏐⏐N θ ⏐⏐⏐ = |a0 |k + Pv−1 av ⏐ , ⏐γn ( p, θ) p k ⏐ ⎭ ⎩ ⏐
(1.2)
v=1
n=1
where 1/k ′
θn pn γ0 ( p, θ) = γn ( p, θ) = , n ≥ 1. Pn Pn−1 ⏐ ⏐ ⏐ θ⏐ Hence it is clear that a ∈ ⏐N p ⏐ if and only if T (a) ∈ lk , the set of all k -absolutely convergent series, where , 1/k ′ θ0 ,
(1.3)
k
T0 (a) = γ0 ( p, θ)a0 , Tn (a) = γn ( p, θ)
n ∑
Pv−1 av ,
(1.4)
v=1
1/k + 1/˜ k = 1 for k > 1, and 1/˜ k = 0 for k = 1. ⏐ ⏐ ⏐ θ⏐ 2. Representations of functionals on the space ⏐N p ⏐
k
It is known that the continuous dual of a normed space X, denoted by X ∗ , is defined by the set of all bounded linear functionals on U, and also it is a fundamental principle of functional analysis that investigations of spaces are often combined with those of the dual spaces. In this connection duals of many spaces have been considered [7]. For example, c∗ ∼ = l1 , l1∗ ∼ = l∞ , lk∗ ∼ = lk ′ for 1 < k < ∞, where c, l∞ and lk ′ are the sets of all convergent, bounded ′ sequences and k -absolutely convergent series, respectively. Also their representations and norms are as follows: f (x) = a lim xn + n
f (x) = f (x) =
∞ ∑
∞ ∑
an xn , ∥ f ∥c∗ = |a| + ∥a∥l1
n=0
an xn , ∥ f ∥l ∗ = ∥a∥∞ (0 < k ≤ 1) k
n=0 ∞ ∑
an xn , ∥ f ∥l ∗ = ∥a∥lk (1 < k < ∞) . k
n=0
⏐ ⏐∗ ⏐ ⏐∗ ⏐ θ⏐ ⏐ θ⏐ In this section showing that ⏐N p ⏐ and ⏐N p ⏐ are isometrically isomorphic to lk ′ and l∞ , respectively, we give general k 1 representations of linear functionals on them and determine ⏐ ⏐ their norms. ⏐ θ⏐ First we characterize the property AK of the space ⏐N p ⏐ , which plays important role to prove the theorems. k
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
M.A. Sarıgöl / Transactions of A. Razmadze Mathematical Institute (
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⏐ ⏐ ⏐ θ⏐ Theorem 2.1. Let 1 ≤ k < ∞ and θ = (θn ) be a sequence of nonnegative numbers. Then, in order that ⏐N p ⏐ is a k B K -space with property AK , it is necessary and sufficient ⏐ ∞ ⏐ ∑ ⏐ γn ( p, θ) ⏐k ⏐ < ∞. ⏐ (2.1) sup ⏐ ⏐ m n=m γm ( p, θ) ⏐ ⏐ ⏐ θ⏐ Proof. ⏐N p ⏐ is a B K -space (see [6]). Now, if x ∈ φ, then there exists a positive integer m such that x = k ⏐ ⏐ ⏐ θ⏐ (x0 , x1 , xm , 0, . . .) , and so φ ⊂ ⏐N p ⏐ iff k
⏐k ∞ ⏐k ⏐ n ⏐ n ⏐ ∑ ⏐∑ ⏐ ∑ ⏐ ⏐ ⏐ ⏐ |γ ( p, θ)|k < ∞, Pv−1 xv ⏐ Pv−1 xv ⏐ = ⏐ ⏐γm ( p, θ) ⏐ m=n m ⏐ ⏐ ⏐ m=n v=1 v=1 ⏐ ⏐ ( (n) ) ⏐ θ⏐ and e is a base of ⏐N p ⏐ iff k m ∑ xn e(n) → 0 as m → ∞. x − ⏐⏐ θ ⏐⏐ ∞ ⏐ ∑
n=0
⏐N p ⏐
k
⏐ ⏐ ⏐ θ⏐ On the other hand, if T (x) ∈ lk for any x ∈ ⏐N p ⏐ , then it follows from Minkowski’s inequality that k
m ∑ (n) xn e x − ⏐⏐ n=0
{ =
θ⏐ ⏐N p ⏐ k
⏐
⏐k } k1 ∞ ⏐ ∑ ⏐ ⏐ T (x) m ⏐ ⏐Tn (x) − γn ( p, θ) → 0 as m → ∞ ⏐ γ ( p, θ) ⏐
n=m+1
m
if and only if ⏐k ⏐ ∞ ⏐ ∞ ⏐ ∑ ∑ ⏐ ⏐ ⏐ γn ( p, θ) ⏐k ⏐γn ( p, θ) Tm (x) ⏐ = |Tm (x)|k ⏐ ⏐ ⏐ ⏐ ⏐ γ ( p, θ) ⏐ → 0 as m → ∞, γ ( p, θ) m m n=m+1 n=m+1 ∑ n or, equivalently, (2.1) holds, which states x = ∞ n=0 x n e . Further, by triangle inequality, it has a unique expression. This proves the result. Theorem 2.2. (i-) Let 1 < k < ∞ and θ = ⏐ (θ⏐n∗) be a sequence of nonnegative numbers.⏐ If ⏐(∗pn ) is a sequence of ⏐ θ⏐ ⏐ θ⏐ nonnegative numbers satisfying (2.1), then, ⏐N p ⏐ is isometrically isomorphic to lk ′ , i.e., ⏐N p ⏐ ∼ = lk ′ . Moreover if k k ⏐ ⏐∗ θ ⏐ ⏐ f ∈ ⏐N p ⏐ k (∞ ) ∞ ⏐ ⏐ ∑ ∑ ⏐ θ⏐ f (x) = λ0 x0 + λn γn ( p, θ) Pv−1 xv ; x ∈ ⏐N p ⏐ (2.2) v=1
n=v
k
and ∥ f ∥⏐⏐
⏐ θ ⏐∗ ⏐N p ⏐
= ∥λ∥lk ′
(2.3)
k
where λ ∈ lk ′ . ⏐ ⏐∗ ⏐ ⏐∗ ⏐ θ⏐ (ii-) Let k = 1 and supn Pn / pn < ∞. Then, ⏐ N p ⏐1 is isometrically isomorphic to l∞ , i.e., ⏐N p ⏐ ∼ = l∞ , and if 1 ⏐ ⏐∗ ⏐ ⏐ f ∈ N p 1 , then it is defined by (2.2) and ∥ f ∥⏐⏐
⏐ θ ⏐′ ⏐N p ⏐
= ∥λ∥∞
(2.4)
1
where λ ∈ l∞ . Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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⏐ ⏐∗ ⏐ θ⏐ Proof. (i) Define T : lk ′ → ⏐N p ⏐ by T (λ) = f, where f is as in (2.2). Trivially, T is well defined by (2.1), linear k ⏐ ⏐∗ ⏐ θ⏐ and injective. Also, T is surjective. In fact, take f ∈ ⏐N p ⏐ . By Lemma 1.6 in [1] we see that (1.4) defines an isometry k ⏐ ⏐ ⏐ ⏐ { ∑∞ } ⏐ θ⏐ ⏐ θ⏐ k 1/k |x | between ⏐N p ⏐ and lk with respect to the norms (1.2) and ∥x∥lk = . This means that x ∈ ⏐N p ⏐ if and n n=0 k k ⏐ ⏐∗ θ ⏐ ⏐ only if T (x) ∈ lk , and ∥x∥⏐⏐⏐N θ ⏐⏐⏐ = ∥T (x)∥lk . Further, f ∈ ⏐N p ⏐ if and only if F ∈ lk′ , where p k k ⏐ ⏐ ⏐ θ⏐ f (x) = F(T (x)) = F(T ), for all x ∈ ⏐N p ⏐ , k
and also ∥f∥ =
| f (x)| = sup |F(T )| = ∥F∥ .
sup ∥x∥⏐⏐ θ ⏐⏐ =1 ⏐N p ⏐
∥T ∥lk =1
k
It is well known from [6] that lk′ ∼ = lk ′ , which shows that F ∈ lk∗ if and only if there exists λ ∈ lk ′ such that F(T ) =
∞ ∑
λn Tn (x), for all T (x) ∈ lk ,
n=0
∥F∥ = ∥λ∥lk ′ .
(2.5)
⏐ ⏐ ⏐ θ⏐ So it follows that for every x ∈ ⏐N p ⏐ f (x) = λ0 x0 +
∞ ∑
λn γn ( p, θ)
k n ∑
Pv−1 xv .
(2.6)
v=1
n=1
To get∑ (2.2), it is sufficient to show that the order of summation in (2.6) can be interchanged. Now, by (2.1), since the series ∞ n=v λn γn ( p, θ) is convergent, we write this sum as f (x) = λ0 x0 + lim
K →∞
= λ0 x0 + lim
K →∞
= λ0 x0 + lim
K →∞
K ∑ n=1 K ∑ v=1 K ∑
λn γn ( p, θ)
n ∑
Pv−1 xv
v=1
Pv−1 xv Pv−1 xv
v=1
K ∑
λn γn ( p, θ).
n=v {∞ ∑
∞ ∑
−
n=v
} λn γn ( p, θ).
n=K +1
Thus it remains to show that ⏐ K ⏐ ∞ ⏐∑ ⏐ ∑ ⏐ ⏐ Pv−1 xv λn γn ⏐ → 0 as K → ∞. ⏐ ⏐ ⏐ v=1
n=K +1
But, it is easily seen from Hölder’s inequality and (2.1) that ⏐ K ⏐ ⏐ ∞ ∞ ⏐ ⏐∑ ⏐ ∑ ∑ ⏐ γn ⏐ ⏐ ⏐ ⏐ λn ⏐ Pv−1 xv λn γn ⏐ ≤ |TK (x)| ⏐ ⏐γ ⏐ ⏐ ⏐ K v=1 n=K +1 n=K +1 ( ∞ ) 1′ k ∑ k′ |λn | ≤ M → 0 as K → ∞ n=K +1
where ⏐ )1 ∞ ⏐ ∑ ⏐ γn ( p, θ) ⏐k k ⏐ ⏐ M = sup |TK (x)| . ⏐ γ ( p, θ) ⏐ K K n=K +1 (
Thus (2.2) holds, and also ∥L(λ)∥⏐⏐
⏐ θ ⏐∗ ⏐N p ⏐ k
= ∥ f ∥⏐⏐
⏐ θ ⏐∗ ⏐N p ⏐
= ∥λ∥lk ′ by (2.5), which completes the proof.
k
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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)
–
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The proof of part (ii) follows from lines in part (i) considering that l1∗ ∼ = l∞ and (2.1) reduces to supn (Pn / pn ) < ∞. Also, using Theorem 2.2, we give a general representation of adjoint operator. We first recall related concepts. Let X, Y be normed spaces and A : X → Y be a bounded linear operator. Then, adjoint operator of A, denoted by A∗ , is defined A∗ : Y ∗ → X ∗ such that A∗ ( f ) = f o A. Making use of Theorem 2.2 ⏐ we ⏐ can prove the following theorem which establishes representation of adjoint ⏐ θ⏐ operator of matrix operator on ⏐N p ⏐ for k ≥ 1. k
Theorem 2.3. Let (( pn ) and sup P / p < ∞ and (2.1), (q ) be sequences of nonnegative numbers⏐ satisfying ⏐ ⏐ ⏐ ⏐⏐ θ ⏐⏐ n) ⏐ ⏐∗n n n ⏐ θ ⏐∗ respectively. If A ∈ ⏐ N p ⏐ , ⏐N q ⏐ , k ≥ 1, then the adjoint operator A∗ : ⏐N q ⏐ → ⏐ N p ⏐ is defined by k
g(x) = A∗ ( f )(x) =
∞ ∑
k
⏐ ⏐ µ j x j ; x ∈ ⏐N q ⏐
j=0
where λ ∈ lk ′ , µ ∈ ℓ∞ and −1/k ′
θj
µ0 = ε0 a00 , µ j =
pj
∞ ∑
∞ ( )∑ Q v−1 P j av j − P j−1 av, j+1
v=1
n=v
λn qn , j ≥ 1. Q n Q n−1
⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ θ⏐ ⏐ θ⏐ Proof. Since ⏐N q ⏐ is a BK-space, by Banach–Steinhaus theorem, A : ⏐ N p ⏐ → ⏐N q ⏐ is a bounded linear operator. k ⏐ ⏐∗ ⏐ k ⏐∗ ⏐ θ⏐ Now, given f ∈ ⏐ N q ⏐ . Then g ∈ ⏐N p ⏐ . So, by Theorem 2.2, there exist λ ∈ l∞ and µ ∈ ℓk ′ such that k
f (x) = λ0 x0 +
(∞ ∞ ∑ ∑ v=1
) ⏐ ⏐ λn γn (q, 1) Q v−1 xv ; x ∈ ⏐ N q ⏐
n=v
and g(x) = µ0 x0 +
(∞ ∞ ∑ ∑ v=1
)
⏐ ⏐ ⏐ θ⏐ µn γn ( p, θ) Pv−1 xv ; x ∈ ⏐N p ⏐ . k
n=v
Also, by g(x) = f (A(x)), g(x) = λ0 A0 (x) +
(∞ ∞ ∑ ∑ v=1
=
∞ ∑ ∞ ∑
) λn γn (q, 1) Q v−1 Av (x)
n=v
εv av j x j ,
v=0 j=0
where ε0 = λ0 , εv = Q v−1
∞ ∑
λn γn (q, 1), v ≥ 1.
n=v
⏐ ⏐ Now if we put x = e( j) ∈ ⏐ N p ⏐ for j = 0, 1, . . . , then we have µ0 = ε0 a00 , P j−1
∞ ∑ n= j
µn γn ( p, θ) =
∞ ∑
εv av j = A Tj (ε)
v=0
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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M.A. Sarıgöl / Transactions of A. Razmadze Mathematical Institute (
)
–
where A T is the transpose of the matrix A. This implies that ( T ) A Tj+1 (ε) A j (ε) 1 µj = − γ j ( p, θ) P j−1 Pj −1/k ′
=
=
θj
pj −1/k ′ θj
pj
∞ ∑
∞ ( )∑ Q v−1 P j av j − P j−1 av, j+1 λn γn (q, 1)
v=1
n=v
∞ ∑
∞ ( )∑ Q v−1 P j av j − P j−1 av, j+1
v=1
n=v
λn q n Q n Q n−1
which completes the proof. Also, following the lines in Theorem 2.4 we get the following theorem. Theorem 2.4. Let (⏐ ( pn ) ⏐and (qn)) be sequences of nonnegative numbers satisfying ⏐ sup ⏐ n Pn / pn < ∞ and (2.1), ⏐ ⏐∗ ⏐ θ ⏐∗ ⏐ θ ⏐ ⏐⏐ ⏐⏐ ∗ ⏐ ⏐ respectively. If A ∈ ⏐N q ⏐ , N p , k > 1, then the adjoint operator A : N p → ⏐N q ⏐ is defined by k
g(x) = A∗ ( f )(x) =
∞ ∑ j=0
k
⏐ ⏐ ⏐ θ⏐ µ j x j ; x ∈ ⏐N q ⏐
k
where λ ∈ lk ′ , µ ∈ ℓ∞ and ′
µ0 = ε0 a00 , µ j =
∞ ∞ 1/k ( )∑ 1 ∑ λn θn qn Q v−1 P j av j − P j−1 av, j+1 , j ≥ 1. p j v=1 Q n Q n−1 n=v
Conflict of interest No conflict of interest was declared by the author. References [1] [2] [3] [4] [5] [6] [7]
M.A. Sarıgöl, On the local properties of factored Fourier series, Appl. Math. Comput. 216 (2010) 3386–3390. H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985) 147–149. W.T. Sulaiman, On summability factors of infinite series, Proc. Amer. Math. Soc. 115 (1992) 313–317. T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc. 7 (1957) 113–141. M.A. Sarıgöl, Matrix transformations on fields of absolute weighted mean summability, Studia Sci. Math. Hungar. 48 (3) (2011) 331–341. M.A. Sarıgöl, Norms and compactness of operators on absolute weighted mean summable series, Kuwait J. Sci. 43 (4) (2016) 68–74. I.J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, 1970.
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
ScienceDirect Transactions of A. Razmadze Mathematical Institute (
)
– www.elsevier.com/locate/trmi
Original article
Representations of functionals on absolute weighted spaces and adjoint operators Mehmet Ali Sarıgöl Department of Mathematics, Pamukkale University, Denizli, Turkey
Received 12 May 2017; received in revised form 8 September 2017; accepted 5 October 2017 Available online xxxx
Abstract In the present paper, we establish general representations of continuous linear functionals, which play important roles in Functional Analysis, of the absolute weighted spaces which have recently been introduced in Sarıgöl (2016, 2011), and also determine their norms. Further making use of this we give adjoint operators of matrix mappings defined on these spaces. c 2017 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V. This is an open access article under the CC ⃝ BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Sequence spaces; BK spaces; Absolute summability; Continuous linear functional
1. Introduction Any vector subspace of w, the space of all (real- or) complex valued sequences, is called a sequence space. A sequence space X is a B K -space if it is a Banach space provided that each of the maps pn : X → C defined by pn (x) = xn is continuous for all n ≥ 0. A B K -space X is said to have AK property if φ ⊂ X and {e(n) } is a basis for X , where e(n) is a sequence whose only non-zero term is 1 in kth place for each n ≥ 0 and φ =span{e(n) }, the set of all finitely non-zero sequences. For example, ℓk , the space of all k-absolutely convergent series, is AK -space for k ≥ 1. Let X, Y be sequence spaces and A = (anv ) be an infinite matrix of complex numbers. If Ax = (An (x)) ∈ Y for every x ∈ ∑ X, then we say that A defines a matrix transformation from X into Y, and denote it by A ∈ (X, Y ), where An (x) = ∞ v=0 anv x v , provided that the series converges for n ≥ 0. Now, let Σ av be a given infinite series with nth partial sums (sn ) and (θn ) be a sequence of nonnegative terms. Then the series Σ av is said to be summable |A, θn |k , k ≥ 1, if ∞ ∑
θnk−1 |∆An (s)|k < ∞, A−1 (s) = 0,
n=0
E-mail address: [email protected]. Peer review under responsibility of Journal Transactions of A. Razmadze Mathematical Institute. https://doi.org/10.1016/j.trmi.2017.10.003 c 2017 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND 2346-8092/⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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)
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∆An (s) = An (s) − An−1 (s) [1].⏐ If A is the⏐ matrix of⏐ weighted mean N , pn (r esp. θn = Pn / pn ) , then summability ⏐ |A, θn |k reduces to summability ⏐ N , pn , θn ⏐k (r esp. ⏐ N , pn ⏐k , [2]), [3]. Further, if θn = n for n ≥ 1 and A is the matrix of Cesàro mean (C, α), then it is the same as summability |C, α|k in Flett’s notation [4]. By a weighted mean matrix we state { pv /Pn , 0 ≤ v ≤ n anv = (1.1) 0, v>n (
)
where ⏐ ⏐ ( pn ) is a sequence of positive numbers with Pn = p0 +⏐ p1 + · · ·⏐ + pn → ∞ as n → ∞. In [5], the space ⏐ θ⏐ ⏐N p ⏐ , k ≥ 1, was defined as the set of all series summable by ⏐ N , pn , θn ⏐k , i.e., k ⎫ ⎧ ⏐k ⏐ n ∞ ⏐ ⎬ ⏐ ⎨ ⏐ ⏐ ∑ ∑ ⏐ ⏐ ⏐ θ⏐ Pv−1 av ⏐ < ∞ , ⏐γn ⏐N p ⏐ = a = (av ) : ⏐ ⎭ ⏐ ⎩ k n=1
v=1
which is a BK-space with respect to the norm (see [6]) ⎧ ⏐k ⎫ k1 ⏐ n ∞ ⏐ ⎨ ⏐ ⎬ ∑ ∑ ⏐ ⏐ ∥a∥⏐⏐⏐N θ ⏐⏐⏐ = |a0 |k + Pv−1 av ⏐ , ⏐γn ( p, θ) p k ⏐ ⎭ ⎩ ⏐
(1.2)
v=1
n=1
where 1/k ′
θn pn γ0 ( p, θ) = γn ( p, θ) = , n ≥ 1. Pn Pn−1 ⏐ ⏐ ⏐ θ⏐ Hence it is clear that a ∈ ⏐N p ⏐ if and only if T (a) ∈ lk , the set of all k -absolutely convergent series, where , 1/k ′ θ0 ,
(1.3)
k
T0 (a) = γ0 ( p, θ)a0 , Tn (a) = γn ( p, θ)
n ∑
Pv−1 av ,
(1.4)
v=1
1/k + 1/˜ k = 1 for k > 1, and 1/˜ k = 0 for k = 1. ⏐ ⏐ ⏐ θ⏐ 2. Representations of functionals on the space ⏐N p ⏐
k
It is known that the continuous dual of a normed space X, denoted by X ∗ , is defined by the set of all bounded linear functionals on U, and also it is a fundamental principle of functional analysis that investigations of spaces are often combined with those of the dual spaces. In this connection duals of many spaces have been considered [7]. For example, c∗ ∼ = l1 , l1∗ ∼ = l∞ , lk∗ ∼ = lk ′ for 1 < k < ∞, where c, l∞ and lk ′ are the sets of all convergent, bounded ′ sequences and k -absolutely convergent series, respectively. Also their representations and norms are as follows: f (x) = a lim xn + n
f (x) = f (x) =
∞ ∑
∞ ∑
an xn , ∥ f ∥c∗ = |a| + ∥a∥l1
n=0
an xn , ∥ f ∥l ∗ = ∥a∥∞ (0 < k ≤ 1) k
n=0 ∞ ∑
an xn , ∥ f ∥l ∗ = ∥a∥lk (1 < k < ∞) . k
n=0
⏐ ⏐∗ ⏐ ⏐∗ ⏐ θ⏐ ⏐ θ⏐ In this section showing that ⏐N p ⏐ and ⏐N p ⏐ are isometrically isomorphic to lk ′ and l∞ , respectively, we give general k 1 representations of linear functionals on them and determine ⏐ ⏐ their norms. ⏐ θ⏐ First we characterize the property AK of the space ⏐N p ⏐ , which plays important role to prove the theorems. k
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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⏐ ⏐ ⏐ θ⏐ Theorem 2.1. Let 1 ≤ k < ∞ and θ = (θn ) be a sequence of nonnegative numbers. Then, in order that ⏐N p ⏐ is a k B K -space with property AK , it is necessary and sufficient ⏐ ∞ ⏐ ∑ ⏐ γn ( p, θ) ⏐k ⏐ < ∞. ⏐ (2.1) sup ⏐ ⏐ m n=m γm ( p, θ) ⏐ ⏐ ⏐ θ⏐ Proof. ⏐N p ⏐ is a B K -space (see [6]). Now, if x ∈ φ, then there exists a positive integer m such that x = k ⏐ ⏐ ⏐ θ⏐ (x0 , x1 , xm , 0, . . .) , and so φ ⊂ ⏐N p ⏐ iff k
⏐k ∞ ⏐k ⏐ n ⏐ n ⏐ ∑ ⏐∑ ⏐ ∑ ⏐ ⏐ ⏐ ⏐ |γ ( p, θ)|k < ∞, Pv−1 xv ⏐ Pv−1 xv ⏐ = ⏐ ⏐γm ( p, θ) ⏐ m=n m ⏐ ⏐ ⏐ m=n v=1 v=1 ⏐ ⏐ ( (n) ) ⏐ θ⏐ and e is a base of ⏐N p ⏐ iff k m ∑ xn e(n) → 0 as m → ∞. x − ⏐⏐ θ ⏐⏐ ∞ ⏐ ∑
n=0
⏐N p ⏐
k
⏐ ⏐ ⏐ θ⏐ On the other hand, if T (x) ∈ lk for any x ∈ ⏐N p ⏐ , then it follows from Minkowski’s inequality that k
m ∑ (n) xn e x − ⏐⏐ n=0
{ =
θ⏐ ⏐N p ⏐ k
⏐
⏐k } k1 ∞ ⏐ ∑ ⏐ ⏐ T (x) m ⏐ ⏐Tn (x) − γn ( p, θ) → 0 as m → ∞ ⏐ γ ( p, θ) ⏐
n=m+1
m
if and only if ⏐k ⏐ ∞ ⏐ ∞ ⏐ ∑ ∑ ⏐ ⏐ ⏐ γn ( p, θ) ⏐k ⏐γn ( p, θ) Tm (x) ⏐ = |Tm (x)|k ⏐ ⏐ ⏐ ⏐ ⏐ γ ( p, θ) ⏐ → 0 as m → ∞, γ ( p, θ) m m n=m+1 n=m+1 ∑ n or, equivalently, (2.1) holds, which states x = ∞ n=0 x n e . Further, by triangle inequality, it has a unique expression. This proves the result. Theorem 2.2. (i-) Let 1 < k < ∞ and θ = ⏐ (θ⏐n∗) be a sequence of nonnegative numbers.⏐ If ⏐(∗pn ) is a sequence of ⏐ θ⏐ ⏐ θ⏐ nonnegative numbers satisfying (2.1), then, ⏐N p ⏐ is isometrically isomorphic to lk ′ , i.e., ⏐N p ⏐ ∼ = lk ′ . Moreover if k k ⏐ ⏐∗ θ ⏐ ⏐ f ∈ ⏐N p ⏐ k (∞ ) ∞ ⏐ ⏐ ∑ ∑ ⏐ θ⏐ f (x) = λ0 x0 + λn γn ( p, θ) Pv−1 xv ; x ∈ ⏐N p ⏐ (2.2) v=1
n=v
k
and ∥ f ∥⏐⏐
⏐ θ ⏐∗ ⏐N p ⏐
= ∥λ∥lk ′
(2.3)
k
where λ ∈ lk ′ . ⏐ ⏐∗ ⏐ ⏐∗ ⏐ θ⏐ (ii-) Let k = 1 and supn Pn / pn < ∞. Then, ⏐ N p ⏐1 is isometrically isomorphic to l∞ , i.e., ⏐N p ⏐ ∼ = l∞ , and if 1 ⏐ ⏐∗ ⏐ ⏐ f ∈ N p 1 , then it is defined by (2.2) and ∥ f ∥⏐⏐
⏐ θ ⏐′ ⏐N p ⏐
= ∥λ∥∞
(2.4)
1
where λ ∈ l∞ . Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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⏐ ⏐∗ ⏐ θ⏐ Proof. (i) Define T : lk ′ → ⏐N p ⏐ by T (λ) = f, where f is as in (2.2). Trivially, T is well defined by (2.1), linear k ⏐ ⏐∗ ⏐ θ⏐ and injective. Also, T is surjective. In fact, take f ∈ ⏐N p ⏐ . By Lemma 1.6 in [1] we see that (1.4) defines an isometry k ⏐ ⏐ ⏐ ⏐ { ∑∞ } ⏐ θ⏐ ⏐ θ⏐ k 1/k |x | between ⏐N p ⏐ and lk with respect to the norms (1.2) and ∥x∥lk = . This means that x ∈ ⏐N p ⏐ if and n n=0 k k ⏐ ⏐∗ θ ⏐ ⏐ only if T (x) ∈ lk , and ∥x∥⏐⏐⏐N θ ⏐⏐⏐ = ∥T (x)∥lk . Further, f ∈ ⏐N p ⏐ if and only if F ∈ lk′ , where p k k ⏐ ⏐ ⏐ θ⏐ f (x) = F(T (x)) = F(T ), for all x ∈ ⏐N p ⏐ , k
and also ∥f∥ =
| f (x)| = sup |F(T )| = ∥F∥ .
sup ∥x∥⏐⏐ θ ⏐⏐ =1 ⏐N p ⏐
∥T ∥lk =1
k
It is well known from [6] that lk′ ∼ = lk ′ , which shows that F ∈ lk∗ if and only if there exists λ ∈ lk ′ such that F(T ) =
∞ ∑
λn Tn (x), for all T (x) ∈ lk ,
n=0
∥F∥ = ∥λ∥lk ′ .
(2.5)
⏐ ⏐ ⏐ θ⏐ So it follows that for every x ∈ ⏐N p ⏐ f (x) = λ0 x0 +
∞ ∑
λn γn ( p, θ)
k n ∑
Pv−1 xv .
(2.6)
v=1
n=1
To get∑ (2.2), it is sufficient to show that the order of summation in (2.6) can be interchanged. Now, by (2.1), since the series ∞ n=v λn γn ( p, θ) is convergent, we write this sum as f (x) = λ0 x0 + lim
K →∞
= λ0 x0 + lim
K →∞
= λ0 x0 + lim
K →∞
K ∑ n=1 K ∑ v=1 K ∑
λn γn ( p, θ)
n ∑
Pv−1 xv
v=1
Pv−1 xv Pv−1 xv
v=1
K ∑
λn γn ( p, θ).
n=v {∞ ∑
∞ ∑
−
n=v
} λn γn ( p, θ).
n=K +1
Thus it remains to show that ⏐ K ⏐ ∞ ⏐∑ ⏐ ∑ ⏐ ⏐ Pv−1 xv λn γn ⏐ → 0 as K → ∞. ⏐ ⏐ ⏐ v=1
n=K +1
But, it is easily seen from Hölder’s inequality and (2.1) that ⏐ K ⏐ ⏐ ∞ ∞ ⏐ ⏐∑ ⏐ ∑ ∑ ⏐ γn ⏐ ⏐ ⏐ ⏐ λn ⏐ Pv−1 xv λn γn ⏐ ≤ |TK (x)| ⏐ ⏐γ ⏐ ⏐ ⏐ K v=1 n=K +1 n=K +1 ( ∞ ) 1′ k ∑ k′ |λn | ≤ M → 0 as K → ∞ n=K +1
where ⏐ )1 ∞ ⏐ ∑ ⏐ γn ( p, θ) ⏐k k ⏐ ⏐ M = sup |TK (x)| . ⏐ γ ( p, θ) ⏐ K K n=K +1 (
Thus (2.2) holds, and also ∥L(λ)∥⏐⏐
⏐ θ ⏐∗ ⏐N p ⏐ k
= ∥ f ∥⏐⏐
⏐ θ ⏐∗ ⏐N p ⏐
= ∥λ∥lk ′ by (2.5), which completes the proof.
k
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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The proof of part (ii) follows from lines in part (i) considering that l1∗ ∼ = l∞ and (2.1) reduces to supn (Pn / pn ) < ∞. Also, using Theorem 2.2, we give a general representation of adjoint operator. We first recall related concepts. Let X, Y be normed spaces and A : X → Y be a bounded linear operator. Then, adjoint operator of A, denoted by A∗ , is defined A∗ : Y ∗ → X ∗ such that A∗ ( f ) = f o A. Making use of Theorem 2.2 ⏐ we ⏐ can prove the following theorem which establishes representation of adjoint ⏐ θ⏐ operator of matrix operator on ⏐N p ⏐ for k ≥ 1. k
Theorem 2.3. Let (( pn ) and sup P / p < ∞ and (2.1), (q ) be sequences of nonnegative numbers⏐ satisfying ⏐ ⏐ ⏐ ⏐⏐ θ ⏐⏐ n) ⏐ ⏐∗n n n ⏐ θ ⏐∗ respectively. If A ∈ ⏐ N p ⏐ , ⏐N q ⏐ , k ≥ 1, then the adjoint operator A∗ : ⏐N q ⏐ → ⏐ N p ⏐ is defined by k
g(x) = A∗ ( f )(x) =
∞ ∑
k
⏐ ⏐ µ j x j ; x ∈ ⏐N q ⏐
j=0
where λ ∈ lk ′ , µ ∈ ℓ∞ and −1/k ′
θj
µ0 = ε0 a00 , µ j =
pj
∞ ∑
∞ ( )∑ Q v−1 P j av j − P j−1 av, j+1
v=1
n=v
λn qn , j ≥ 1. Q n Q n−1
⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ θ⏐ ⏐ θ⏐ Proof. Since ⏐N q ⏐ is a BK-space, by Banach–Steinhaus theorem, A : ⏐ N p ⏐ → ⏐N q ⏐ is a bounded linear operator. k ⏐ ⏐∗ ⏐ k ⏐∗ ⏐ θ⏐ Now, given f ∈ ⏐ N q ⏐ . Then g ∈ ⏐N p ⏐ . So, by Theorem 2.2, there exist λ ∈ l∞ and µ ∈ ℓk ′ such that k
f (x) = λ0 x0 +
(∞ ∞ ∑ ∑ v=1
) ⏐ ⏐ λn γn (q, 1) Q v−1 xv ; x ∈ ⏐ N q ⏐
n=v
and g(x) = µ0 x0 +
(∞ ∞ ∑ ∑ v=1
)
⏐ ⏐ ⏐ θ⏐ µn γn ( p, θ) Pv−1 xv ; x ∈ ⏐N p ⏐ . k
n=v
Also, by g(x) = f (A(x)), g(x) = λ0 A0 (x) +
(∞ ∞ ∑ ∑ v=1
=
∞ ∑ ∞ ∑
) λn γn (q, 1) Q v−1 Av (x)
n=v
εv av j x j ,
v=0 j=0
where ε0 = λ0 , εv = Q v−1
∞ ∑
λn γn (q, 1), v ≥ 1.
n=v
⏐ ⏐ Now if we put x = e( j) ∈ ⏐ N p ⏐ for j = 0, 1, . . . , then we have µ0 = ε0 a00 , P j−1
∞ ∑ n= j
µn γn ( p, θ) =
∞ ∑
εv av j = A Tj (ε)
v=0
Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.
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)
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where A T is the transpose of the matrix A. This implies that ( T ) A Tj+1 (ε) A j (ε) 1 µj = − γ j ( p, θ) P j−1 Pj −1/k ′
=
=
θj
pj −1/k ′ θj
pj
∞ ∑
∞ ( )∑ Q v−1 P j av j − P j−1 av, j+1 λn γn (q, 1)
v=1
n=v
∞ ∑
∞ ( )∑ Q v−1 P j av j − P j−1 av, j+1
v=1
n=v
λn q n Q n Q n−1
which completes the proof. Also, following the lines in Theorem 2.4 we get the following theorem. Theorem 2.4. Let (⏐ ( pn ) ⏐and (qn)) be sequences of nonnegative numbers satisfying ⏐ sup ⏐ n Pn / pn < ∞ and (2.1), ⏐ ⏐∗ ⏐ θ ⏐∗ ⏐ θ ⏐ ⏐⏐ ⏐⏐ ∗ ⏐ ⏐ respectively. If A ∈ ⏐N q ⏐ , N p , k > 1, then the adjoint operator A : N p → ⏐N q ⏐ is defined by k
g(x) = A∗ ( f )(x) =
∞ ∑ j=0
k
⏐ ⏐ ⏐ θ⏐ µ j x j ; x ∈ ⏐N q ⏐
k
where λ ∈ lk ′ , µ ∈ ℓ∞ and ′
µ0 = ε0 a00 , µ j =
∞ ∞ 1/k ( )∑ 1 ∑ λn θn qn Q v−1 P j av j − P j−1 av, j+1 , j ≥ 1. p j v=1 Q n Q n−1 n=v
Conflict of interest No conflict of interest was declared by the author. References [1] [2] [3] [4] [5] [6] [7]
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Please cite this article in press as: M.A. Sarıgöl, Representations of functionals on absolute weighted spaces and adjoint operators, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.10.003.