On Some Euler Sequence Spaces of Nonabsolute Type

In the present paper, the Euler sequence spaces eʳ₀ and eʳc of nonabsolute type which are the BK-spaces including the spaces c₀ and c have been introduced and proved that the spaces er₀ and erᶜ are linearly i somorphic to the spaces c₀ and c, respectively. Furthemore, some inclusion theorems have be...

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Datum:	2005
Hauptverfasser:	Altay, B., Başar, F.
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spelling	irk-123456789-1659192020-02-18T01:27:07Z On Some Euler Sequence Spaces of Nonabsolute Type Altay, B. Başar, F. Статті In the present paper, the Euler sequence spaces eʳ₀ and eʳc of nonabsolute type which are the BK-spaces including the spaces c₀ and c have been introduced and proved that the spaces er₀ and erᶜ are linearly i somorphic to the spaces c₀ and c, respectively. Furthemore, some inclusion theorems have been given. Additionally, the α−,β−,γ− and continuous duals of the spaces eʳ₀ and eʳc have been computed and their basis have been constructed. Finally, the necessary and sufficient conditions on an infinite matrix belonging to the classes (eʳc : lp) and (eʳc : c) have been determined and the characterizations of some other classes of infinite matrices have also been derived by means of a given basic lemma, where 1 ≤ p ≤ ∞. Введено поняття просторів послідовностей Ейлера eʳ₀ та eʳc неабсолютного типу — BK-просторів, що містять простори c₀ та c. Доведено, що простори eʳ₀ та eʳc лінійно ізоморфні відповідно до просторів c₀ та c. Наведено деякі теореми про включення. Крім того, обчислено α−,β−,γ− та неперервні простори, дуальні до просторів eʳ₀ та erc, і побудовано базиси цих просторів. Визначено необхідні та достатні умови належності нескінченної матриці до класів (eʳc : lp) та (eʳc : c). Отримано характеристики деяких інших класів нескінченних матриць з використанням наведеної в роботі основної леми для випадку 1 ≤ p ≤ ∞. 2005 Article On Some Euler Sequence Spaces of Nonabsolute Type / B. Altay, F. Başar // Український математичний журнал. — 2005. — Т. 57, № 1. — С. 3–17. — Бібліогр.: 19 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165919 517.9 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Altay, B. Başar, F. On Some Euler Sequence Spaces of Nonabsolute Type Український математичний журнал
description	In the present paper, the Euler sequence spaces eʳ₀ and eʳc of nonabsolute type which are the BK-spaces including the spaces c₀ and c have been introduced and proved that the spaces er₀ and erᶜ are linearly i somorphic to the spaces c₀ and c, respectively. Furthemore, some inclusion theorems have been given. Additionally, the α−,β−,γ− and continuous duals of the spaces eʳ₀ and eʳc have been computed and their basis have been constructed. Finally, the necessary and sufficient conditions on an infinite matrix belonging to the classes (eʳc : lp) and (eʳc : c) have been determined and the characterizations of some other classes of infinite matrices have also been derived by means of a given basic lemma, where 1 ≤ p ≤ ∞.
format	Article
author	Altay, B. Başar, F.
author_facet	Altay, B. Başar, F.
author_sort	Altay, B.
title	On Some Euler Sequence Spaces of Nonabsolute Type
title_short	On Some Euler Sequence Spaces of Nonabsolute Type
title_full	On Some Euler Sequence Spaces of Nonabsolute Type
title_fullStr	On Some Euler Sequence Spaces of Nonabsolute Type
title_full_unstemmed	On Some Euler Sequence Spaces of Nonabsolute Type
title_sort	on some euler sequence spaces of nonabsolute type
publisher	Інститут математики НАН України
publishDate	2005
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/165919
citation_txt	On Some Euler Sequence Spaces of Nonabsolute Type / B. Altay, F. Başar // Український математичний журнал. — 2005. — Т. 57, № 1. — С. 3–17. — Бібліогр.: 19 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT altayb onsomeeulersequencespacesofnonabsolutetype AT basarf onsomeeulersequencespacesofnonabsolutetype
first_indexed	2025-07-14T19:07:01Z
last_indexed	2025-07-14T19:07:01Z
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fulltext	UDC 517.9 B. Altay, F. Bas,ar (Inönü Üniv., Turkey) SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE DEQKI PROSTORY POSLIDOVNOSTEJ EJLERA NEABSOLGTNOHO TYPU In the present paper, the Euler sequence spaces er 0 and ec r of nonabsolute type which are the BK- spaces including the spaces c0 and c have been introduced and proved that the spaces er 0 and ec r are linearly i somorphic to the spaces c0 and c, respectively. Furthemore, some inclusion theorems have been given. Additionally, the α-, β-, γ- and continuous duals of the spaces er 0 and ec r have been computed and their basis have been constructed. Finally, the necessary and sufficient conditions on an infinite matrix belonging to the classes ec r p: �( ) and e cc r :( ) have been determined and the characterizations of some other classes of infinite matrices have also been derived by means of a given basic lemma, where 1 ≤ p ≤ ∞. Vvedeno ponqttq prostoriv poslidovnostej Ejlera er 0 ta ec r neabsolgtnoho typu — VK-pro- storiv, wo mistqt\ prostory c0 ta c. Dovedeno, wo prostory er 0 ta ec r linijno izomorfni vidpovidno do prostoriv c0 ta c. Navedeno deqki teoremy pro vklgçennq. Krim toho, obçys- leno α-, β-, γ- ta neperervni prostory, dual\ni do prostoriv er 0 ta ec r , i pobudovano bazysy cyx prostoriv. Vyznaçeno neobxidni ta dostatni umovy naleΩnosti neskinçenno] matryci do kla- siv ec r p: �( ) ta e cc r :( ). Otrymano xarakterystyky deqkyx inßyx klasiv neskinçennyx matryc\ z vykorystannqm navedeno] v roboti osnovno] lemy dlq vypadku 1 ≤ p ≤ ∞. 1. Preliminaries, background and notation. By w, we shall denote the space of all real or complex valued sequences. Any vector subspace of w 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; here, 1 < p < ∞. A sequence space λ with a linear topology is called a K-space provided each of the maps pi : λ → C defined by p x xi i( ) = is continuous for all i ∈N ; here, C denotes the complex field and N = …{ , , , }0 1 2 . A K-space λ is called an FK-space provided λ is a complete linear metric space. An FK-space whose topology is normable is called a BK-space (see [1, p. 272, 273]). Let λ , µ be two sequence spaces and A ank= ( ) be an infinite matrix of real or complex numbers ank , where k n, ∈N . Then, we say that A defines a matrix mapping from λ into µ, and we denote it by writing A: λ µ→ , if for every sequence x xk= ∈( ) λ , the sequence Ax Ax n= { }( ) , the A-transform of x, is in µ; here, ( )Ax a xn k nk k= ∑ , n ∈N . (1.1) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By ( : )λ µ , we denote the class of all matrices A such that A : © B. ALTAY, F. BAS, AR , 2005 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 3 4 B. ALTAY, F. BAS, AR 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 l if Ax converges to l which is called as the A-limit of x. The matrix domain λA of an infinite matrix A is a sequence space λ is defined by λ λA kx x w Ax= = ∈ ∈{ }( ) : (1.2) which is a sequence space. The new sequence space λA generated by the limitation matrix A from the space λ either includes the space λ or is included by the space λ, in general, i.e., the space λA is the expansion of the contraction of the original space λ. The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has been employed by Wang [2], Ng and Lee [3], Malkowsky [4], and Bas,ar and Altay [5]. They introduced the sequence spaces �p Nq ( ) in [2], �∞ ∞( ) =C X 1 and �p C pX( ) = 1 in [3], �∞ ∞( ) =R t t r , c r R c t t = and c rR t t0 0( ) = in [4] and �p pb( ) = ∆ v in [5]; here, Nq , C1 and Rt denote the Nörlund, arithmetic and Riesz means, respectively, and ∆ also denotes the band matrix defining the difference operator and 1 ≤ p < ∞. Quite recently, Aydın and Bas,ar have studied the sequence spaces c aA r r0 0( ) = , c a A c r r = in [6], �p A p r r a( ) = , �∞ ∞( ) =A r r a in [7], a ar r 0 0( )∆ ∆ = ( ) , a ac r c r( )∆ ∆ = ( ) in [8] and extended the sequence spaces ar 0, ac r to the paranormed spaces a pr 0( ), a pc r( ) in [9]; here, Ar denotes the matrix A ar nk r= ( ) defined by a r n k n k n nk r k = + + ≤ ≤ >      1 1 0 0 , , , , for all k n, ∈N and any fixed r ∈R . In the present paper, following [2 – 9], we introduce the Euler sequence spaces er 0 and ec r of nonabsolute type and derive some results related to those sequence spaces. Furthermore, we have constructed the basis and computed the α-, β-, γ-, and continuous duals of the spaces er 0 and ec r . Finally, we have essentially characterized the matrix classes ec r p: �( ) , e cc r :( ) and also derived the characterizations of some other classes by means of a given basic lemma, where 1 ≤ p ≤ ∞. Besides, we have stated and proved a Steinhaus type theorem concerning with the disjointness of the classes e cr ∞( ): and e cc r s :( ) . 2. The Euler sequence spaces er 0 and ec r of nonabsolute type. Altay, Bas,ar and Mursaleen [10] have recently studied Euler sequence spaces ep r and er ∞ consisting of the sequences whose Er -transforms are in the spaces �p and �∞ , respectively; here, 1 ≤ p < ∞ and Er denotes the Euler means of order r defined by the matrix E er nk r= ( ), enk r = n k r r k n k n n k k    − ≤ ≤ >     −( ) , , , , 1 0 0 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 5 for all k n, ∈N . It is known that the method Er is regular for 0 < r < 1 (see [11, p. 57]), and we assume unless stated otherwise that 0 < r < 1. Mursaleen, Bas,ar and Altay have given the characterizations of the matrix classes related to the spaces ep r and er ∞ , and emphasized some geometric properties, for example, Banach – Saks and weak Banach – Saks properties, etc., of the space ep r in [12]. Continuing on this way, we introduce the Euler sequence spaces e x x w n k r r xr k n k n n k k k0 0 1 0= = ( ) ∈     − =        →∞ = −∑: lim ( ) and e x x w n k r r xc r k n k n n k k k= = ( ) ∈     −        →∞ = −∑: lim ( ) 0 1 exists . With the notation of (1.2), we may redefine the spaces er 0 and ec r as follows: e cr Er0 0= ( ) and e cc r Er= . (2.1) It is trivial that e er c r 0 ⊂ . If λ is any normed sequence space, then we call the matrix domain λ Er as the Euler sequence space. Define the sequence y y rk= { }( ) , which will be frequently used, as the Er -transform of a sequence x xk= ( ), i.e., y r k j r r xk j k k j j j( ) ( )=     − = −∑ 0 1 , k ∈N . (2.2) We now may begin with the following theorem which is essential in the text: Theorem 2.1. The sets er 0 and ec r are the linear spaces with the coordinatewise addition and scalar multiplication which are the BK-spaces with the norm x er 0 = = x ec r = E xr �∞ . Proof. The first part of the theorem is a routine verification and so we omit it. Furthermore, since (2.1) holds and c0 , c are the BK-spaces with respect to their natural norm (see [13, p. 217, 218]), and the matrix E er nk r= ( ) is normal, i.e., enn r ≠ 0 , enk r = 0, k > n, for all k n, ∈N , Theorem 4.3.2 of Wilansky [14, p. 61] gives the fact that the spaces er 0, ec r are the BK-spaces. The theorem is proved. Therefore, one can easily check that the absolute property does not hold on the spaces er 0 and ec r , since x xe e r r 0 0 ≠ and x xe ec r c r≠ for at least one sequence in the spaces er 0 and ec r , where x xk= ( ) . This says that er 0 and ec r are the sequence spaces of nonabsolute type. Theorem 2.2. The Euler sequence spaces er 0 and ec r of nonabsolute type are linearly isomorphic to the spaces c0 and c, respectively, i.e., e cr 0 0≅ and e cc r ≅ . Proof. To prove this, we should show the existence of a linear bijection between the spaces er 0 and c0 . Consider the transformation T defined, with the notation of (2.2), from er 0 to c0 by x y Tx� = . The linearity of T is clear. Further, it is trivial that x = θ whenever Tx = θ and hence T is injective, where θ = (0, 0, 0, … ). ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 6 B. ALTAY, F. BAS, AR Let y c∈ 0 and define the sequence x x rk= { }( ) by x r k j r r yk j k k j k j( ) ( )=     − = − −∑ 0 1 , k ∈N . Then, we have lim lim ( ) ( ) lim n r n n k n n k k j k k j k j n nE x n k r r k j r r y y →∞ →∞ = − = − − → ∞( ) =     −     −       = =∑ ∑ 0 0 1 1 0 which says us that x er∈ 0 . Additionally, we observe that x er 0 = sup ( ) ( ) n k n n k k j k k j k j n k r r k j r r y ∈ = − = − −∑ ∑    −     − N 0 0 1 1 = = sup n n cy y ∈ = < ∞ N 0 . Consequently, we see from here that T is surjective and is norm preserving. Hence, T is a linear bijection which therefore says us that the spaces er 0 and c0 are linearly isomorphic, as was desired. It is clear here that if the spaces er 0 and c0 are respectively replaced by the spaces ec r and c, then we obtain the fact that e cc r ≅ . This completes the proof. We now may give our two theorems on the inclusion relations concerning with the spaces er 0 and ec r . Theorem 2.3. Although the inclusions c er 0 0⊂ and c ec r⊂ strictly hold, neither of the spaces er 0 and �∞ includes the other one. Proof. Let us take any s c∈ 0. Then, bearing in mind the regularity of the Euler means of order r , we immediately observe that E s cr ∈ 0 which means that s er∈ 0 . Hence, the inclusion c er 0 0⊂ holds. Furthermore, let us consider the sequence u u rk= { }( ) defined by u r rk k( ) = −( )− for all k ∈N . Then, since E ur = = ( )−{ }r k ∈ c0 , u is in er 0 but not in c0 . By the similar discussion, one can see that the inclusion c ec r⊂ also holds. To establish the second part of theorem, consider that sequence u u rk= { }( ) defined above, and x = e = (1, 1, 1, … ). Then, u is in er 0 but not in �∞ and x is in �∞ but not in er 0. Hence, the sequence spaces er 0 and �∞ overlap but neither contains the other. This completes the proof. Theorem 2.4. If 0 < t < r < 1, then e er t 0 0⊂ and e ec r c t⊂ . Proof. Let us take x x ek r= ( ) ∈ 0. Then, for all k ∈N , we observe that z e x e e y e yk i k ki t i i k ki t j i ij r j j k kj t r j= =       = = = = = ∑ ∑ ∑ ∑ 0 0 0 1 0 / / . Since 0 < t r < 1, the method Et r/ is regular which implies that z z ck= ( ) ∈ 0 whenever y y ck= ( ) ∈ 0 and we thus see that x x ek t= ( ) ∈ 0. This means that the inclusion e er t 0 0⊂ holds. ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 7 Now, one can show in the similar way that the inclusion e ec r c t⊂ also holds and so we leave the detail to the reader. 3. The basis for the spaces er 0 and ec r . In the present section, we give two sequences of the points of the spaces er 0 and ec r which form the basis for the spaces er 0 and ec r . Firstly, we define the Schauder basis of a normed space. If a normed sequence space λ contains a sequence bn( ) with the property that, for every x ∈λ , there is a unique sequence of scalars αn( ) such that lim n n nx b b b →∞ − + +…+( ) =α α α0 0 1 1 0 , then bn( ) is called a Schauder basis (or briefly basis) for λ. The series αk kb∑ which has the sum x is then called the expansion of x with respect to bn( ), and written as x bk k= ∑α . Theorem 3.1. Define the sequence b r b rk n k n ( ) ( )( ) ( )= { } ∈N of the elements of the space er 0 by b r n k n k r r n k n k n k n ( )( ) , , ( ) , , = ≤ <     − ≥     − − 0 0 1 (3.1) for every fixed k ∈N . Then: (i) The sequence b rk k ( )( ){ } ∈N is a basis for the space er 0 and any x er∈ 0 has a unique representation of the form x r b r k k k= ∑λ ( ) ( )( ) . (3.2) (ii) The set e b rk, ( )( ){ } is a basis for the space ec r and any x ec r∈ has a unique representation of the form x le r l b r k k k= + −[ ]∑ λ ( ) ( )( ) , (3.3) where λk r k r E x( ) = ( ) for all k ∈N and l E x k r k = ( ) →∞ lim . (3.4) Proof. (i) It is clear that b r ek r( )( ){ } ⊂ 0 , since E b r e cr k k( ) ( )( ) = ∈ 0 , k = 0, 1, 2, … , (3.5) where e k( ) is the sequence whose only nonzero term is a 1 in k-th place for each k ∈N . Let x er∈ .0 be given. For every nonnegative integer m, we put x r b rm k m k k[ ] ( )( ) ( )= = ∑ 0 λ . (3.6) ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 8 B. ALTAY, F. BAS, AR Then, by applying Er to (3.6) with (3.5), we obtain that E x r E b r E x er m k m k r k k m r k k[ ] ( ) ( )( ) ( )= = ( ) = = ∑ ∑ 0 0 λ and E x x i m E x i m r m i r i −( ){ } = ≤ ≤ ( ) >     [ ] , , , , 0 0 i m, ∈N . Given ε > 0, then there is an integer m0 such that E xr m( ) < ε 2 for all m m≥ 0. Hence, x x E x E xm e n m r n n m r nr− = ( ) ≤ ( ) ≤ < ≥ ≥ [ ] sup sup 0 0 2 ε ε for all m m≥ 0 which proves that x er∈ 0 is represented as in (3.2). Let us show the uniqueness of the representation for x er∈ 0 given by (3.2). Suppose, on the contrary, that there exists a representation x r b r k k k= ∑ µ ( ) ( )( ) . Since the linear transformation T, from er 0 to c0 , used in the proof of Theorem 2.2 is continuous, at this stage we have E x r E b r r e rr n k k r k n k k n k n( ) = { } = =∑ ∑µ µ µ( ) ( ) ( ) ( )( ) ( ) , n ∈N , which contradicts the fact that E x rr n n( ) = λ ( ) for all n ∈N . Hence, the representation (3.2) of x er∈ 0 is unique. Thus, the proof of the first part of theorem is completed. (ii) Since b r ek r( )( ){ } ⊂ 0 and e c∈ , the inclusion e b r ek c r, ( )( ){ } ⊂ trivially holds. Let us take x ec r∈ . Then, there uniquely exists an l satisfying (3.4). We thus have the fact that u er∈ 0 whenever we set u = x – le. Therefore, we deduce by the part (i) of the present theorem that the representation of u is unique. This leads us to the fact that the representation of x given by (3.3) is unique and this step concludes the proof. 4. The αααα -, ββββ-, γγγγ- and continuous duals of the spaces er 0 and ec r . In this section, we state and prove the theorems determining the α-, β-, γ - and continuous duals of the sequence spaces er 0 and ec r of nonabsolute type. For the sequence spaces λ and µ, define the set S λ µ,( ) by S z z w xz x z xk k kλ µ µ λ, :( ) = = ( ) ∈ = ( ) ∈ ∈{ }for all . (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 λ λα = ( )S , �1 , λ λβ = ( )S cs, , and λ λγ = ( )S bs, . It is well known that ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 9 � �p q( ) =β and � �∞( ) =β 1, (4.2) where 1 ≤ p < ∞ and p q− −+ =1 1 1 (see [15, p. 68, 69]). We shall throughout denote the collection of all finite subsets of N by F. The continuous dual of a normed space X is defined as the space of all bounded linear functionals on X and is denoted by X. We shall begin with quoting the lemmas, due to Stieglitz and Tietz [16], which are needed in proving Theorems 4.1 – 4.3. Lemma 4.1. A c c∈( ) = ( )0 1 1: :� � if and only if sup K n k K nka ∈ ∈ ∑ ∑ < ∞ F . Lemma 4.2. A c c∈( )0 : if and only if lim n nk ka →∞ = α , k ∈N , (4.3) sup n k nka ∈ ∑ < ∞ N . (4.4) Lemma 4.3. A c∈( )∞0 : � if and only if (4.4) holds. Theorem 4.1. The α-dual of the spaces er 0 and ec r is b a a w n k r r ar k K n k K n k n n= = ( ) ∈     − < ∞        ∈ ∈ − −∑ ∑: sup ( ) F 1 . Proof. Let a a wn= ( ) ∈ and define the matrix Br whose rows are the product of the rows of the matrix E r1/ and the sequence a an= ( ) . Bearing in mind the relation (2.2), we immediately derive that a x n k r r a y B yn n k n n k n n k r n =     − = ( ) = − −∑ 0 1( ) , n ∈N . (4.5) We therefore observe by (4.5) that ax a xn n= ( ) ∈�1 whenever x er∈ 0 or ec r if and only if B yr ∈�1 whenever y c∈ 0 of c. Then we derive by Lemma 4.1 that sup ( ) K n k K n k n n n k r r a ∈ ∈ − −∑ ∑     − < ∞ F 1 which yields the consequence that e e br c r r0{ } = { } = α α . Theorem 4.2. Define the sets dr 1 , dr 2 , and dr 3 by d a a w j k r r ar k n k n j k n j k j j1 0 1= = ( ) ∈     − < ∞        ∈ = = − −∑ ∑: sup ( ) N , d a a w j k r r a kr k j k j k j j2 1= = ( ) ∈     − ∈        = ∞ − −∑: ( ) exists for each N , and ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 10 B. ALTAY, F. BAS, AR d a a w j k r r ar k n k n j k n j k j j3 0 1= = ( ) ∈     −        →∞ = = − −∑ ∑: lim ( ) exists . Then e d dr r r 0 1 2{ } = β ∩ and e d d dc r r r r{ } = β 1 2 3∩ ∩ . Proof. Because of the proof may also be obtained for the space ec r in the similar way, we omit it and give the proof only for the space er 0. Consider the equation k n k ka x = ∑ 0 = k n j k k j k j k k j r r y a = = − −∑ ∑     −        0 0 1( ) = = k n j k n j k j j k j k r r a y = = − −∑ ∑     −        0 1( ) = T yr n( ) , (4.6) where T tr nk r= ( ) is defined by t j k r r a k n k n nk r j k n j k j j=     − ≤ ≤ >      = − −∑ ( ) , , , , 1 0 0 k n, ∈N . (4.7) Thus, we deduce from Lemma 4.2 with (4.6) that ax a x csx k= ( ) ∈ whenever x = = x ek r( ) ∈ 0 if and only if T y cr ∈ whenever y y ck= ∈( ) 0 . Therefore, we derive from (4.3) and (4.4) that lim n nk rt →∞ exists for each k ∈N and sup n k n nk rt ∈ = ∑ < ∞ N 0 (4.8) which shows that e d dr r r 0 1 2{ } = β ∩ . Theorem 4.3. The γ-dual of the spaces er 0 and ec r is dr 1 . Proof. It is of course that the present theorem may be proved by the technique used in the proof of Theorems 4.1 and 4.2, above. But we prefer here following the classical way and give the proof for the space er 0. Let a a dk r= ( ) ∈ 1 and x x ek r= ( ) ∈ 0. Consider the following equality: k n k k k n k j k k j k j k n nk r k k n nk r ka x a k j r r y t y t y = = = − − = = ∑ ∑ ∑ ∑ ∑=     −         = ≤ 0 0 0 0 0 1( ) which gives us by taking supremum over n ∈N that sup sup sup n k n k k n k n nk r k c n k n nk ra x t y y t ∈ = ∈ = ∈ = ∑ ∑ ∑≤     ≤     ≤ ∞ N N N0 0 0 0 . This means that a a ek r= ( ) ∈{ }0 γ . Hence, d er r 1 0⊂ { }γ . (4.9) Conversely, let a a ek r= ( ) ∈{ }0 γ and x er∈ 0 . Then, one can easily see that k n nk r k n t y= ∈ ∞∑{ } ∈ 0 N � whenever a x bsk k( ) ∈ . This shows that the triangle matrix T tr nk r= ( ), defined by (4.7), is in the class c0 : �∞( ) . Hence, the condition ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 11 sup n k n nk rt ∈ = ∑ < ∞ N 0 is satisfied which yields that a a dk r= ( ) ∈ 1. That is to say that e dr r 0 1{ } ⊂ γ . (4.10) Therefore by combining the inclusions (4.9) and (4.10), we deduce that the γ-dual of the space er 0 is dr 1 and this completes the proof. Theorem 4.4. ec r{ } and er 0{ }* are isometrically isomorphic to �1. Proof. We only give the proof for the space ec r . Suppose that f ec r∈{ }* . Since by the part (ii) of Theorem 3.1, e b rk, ( ){ } is a basis for the space ec r and any element x ec r∈ can be expressed as in the form of (3.3). By the linearity and the continuity of f, we get from (3.3) that f x lf e r l f b r k k k( ) ( ) ( ) ( )( )= + −[ ] { }∑ λ for all x ec r∈ . Define the sequence x x r ek c r= { } ∈( ) such that x ec r = 1 by x r k j r r f b r k n k j r r f b r k n k j k k j k j j n k j k j ( ) ( ) sgn ( ) , , ( ) sgn ( ) , . ( ) ( ) =     − ( ) ≤ ≤     − ( ) >        = − − = − − ∑ ∑ 0 0 1 0 1 Therefore, we have f x f b r f k n k( ) ( )( )= ( ) ≤ = ∑ 0 . (4.11) It follows from the inequality (4.11) that k k n k n kf b r f b r f∑ ∑( ) = ( ) ≤ ∈ = ( ) ( )( ) sup ( ) N 0 . Write f x al a r k k k( ) ( )= + ∑ λ , where a f e f b r k k= − ( )∑( ) ( )( ) , a f b rk k= ( )( )( ) , the series k kf b r∑ ( )( )( ) being absolutely convergent. Since limk r k E x→∞( ) ≤ ≤ x ec r , we have f x x a ae k kc r( ) ≤ +    ∑ , whence f a a k k≤ + ∑ . (4.12) Also, for x ec r = 1, we have ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 12 B. ALTAY, F. BAS, AR f x f( ) ≤ , so we define for any n ≥ 0, x r k j r r a k n k j r r a k j r r a k n k j k k j k j j n k j k j j n k k j k ( ) ( ) sgn , , ( ) sgn ( ) sgn , . =     − ≤ ≤     − +     − >        = − − = − − = + − − ∑ ∑ ∑ 0 0 1 1 0 1 1 Then x ec r∈ with x ec r = 1, lim sgnk r k E x a→∞( ) = and so f x a a a a f k n k k n k( ) sgn= + + ≤ = = + ∞ ∑ ∑ 0 1 . (4.13) Since ( )ak ∈�1 we have k n ka= + ∞∑ → 1 0 as n →∞ , and thus we obtain by letting n →∞ in (4.13) that a a f k k+ ≤∑ . (4.14) Combining the results (4.12) and (4.14) we see that f a a k k= + ∑ which is the norm on �1. Now, let T ec r: *{ } → �1 be defined by f a a a� , , ,0 1 …( ) . Then, we have T f a a a f( ) = + + +… =0 1 . T f( ) being the �1-norm. Thus, T is norm preserving. T is obviously surjective and linear, and hence is an isomorphism from ec r{ }γ to �1. This completes the proof. 5. Some matrix mappings related to the Euler sequence spaces. In this section, we characterize the matrix mappings from ec r into some of the known sequence spaces and into the Euler, difference, Riesz, Cesàro sequence spaces. We directly prove the theorems characterizing the classes ec r p: �( ) , e cc r :( ) and derive the other characterizations from them by means of a given basic lemma, where 1 ≤ p ≤ ∞ . Furthermore, we give a Steinhaus-type theorem which asserts that the classes e cr ∞( ): and e cc r s :( ) are disjoint. We shall write throughout for brevity that a n k a j n jk( , ) = = ∑ 0 and ˜ ( )a j k r r ank j k n j k j nj=     − = − −∑ 1 for all k, n ∈N . We will also use the similar notations with other letters and use the convention that any term with negative subscript is equal to naught. We shall begin with two lemmas which are needed in the proof of our theorems. Lemma 5.1 [14, p. 57]. The matrix mappings between the BK-spaces are continuous. ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 13 Lemma 5.2 [14, p. 128]. A c p∈( ): � if and only if sup F n k F nk p a ∈ ∈ ∑ ∑ < ∞ F , 1 ≤ p < ∞. (5.1) Theorem 5.1. A ec r p∈( ): � if and only if (i) for 1 ≤ p < ∞, sup ˜ F n k F nk p a ∈ ∈ ∑ ∑ < ∞ F , (5.2) ãnk exists for all k n, ∈N , (5.3) k nka∑ ˜ converges for all n ∈N , (5.4) sup ( ) m k m j k m j k j nj j k r r a ∈ = = − −∑ ∑     − < ∞ N 0 1 , n ∈N; (5.5) (ii) for p = ∞, (5.3) and (5.5) hold, and sup ˜ n k nka ∈ ∑ < ∞ N . (5.6) Proof. Suppose conditions (5.2) – (5.5) hold and take any x ec r∈ . Then, a enk k c r{ } ∈{ }∈N β for all n ∈N and this implies that Ax exists. Let us define the matrix B bnk= ( ) with b ank nk= ˜ for all k n, ∈N . Then, since (5.1) is satisfied for that matrix B , we have B c p∈( ): � . Let us now consider the following equality obtained from the m-th partial sum of the series k nk ka x∑ : k m nk k k m j k m j k j nj ka x j k r r a y = = = − −∑ ∑ ∑=     − 0 0 1( ) , m n, ∈N. (5.7) Therefore, we derive from (5.7) as m → ∞ that k nk k k nk ka x a y∑ ∑= ˜ , n ∈N , (5.8) which yields by taking �p -norm that Ax By p p� �= < ∞ . This means that A ec r p∈( ): � . Conversely, suppose that A ec r p∈( ): � . Then, since ec r and �p are the BK-spaces, we have from Lemma 5.1 that there exists some real constant K > 0 such that Ax K x p c re� ≤ (5.9) for all x ec r∈ . Since inequality (5.9) also holds for the sequence x = xk( ) = = k F kb r∈∑ ( )( ) belonging to the space ec r , where b r b rk n k( ) ( )( ) ( )= { } is defined by (3.1), we thus have for any F ∈F that ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 14 B. ALTAY, F. BAS, AR Ax a K x K p c r n k F nk p p e� =       ≤ =∑ ∑ ∈ ˜ /1 which shows the necessity of (5.2). Since A is applicable to the space ec r by the hypothesis, the necessity of conditions (5.3) – (5.5) is trivial. This completes the proof of the part (i) of theorem. Since the part (ii) may also be proved in the similar way that of the part (i), we leave the detailed proof to the reader. Theorem 5.2. A e cc r∈( ): if and only if (5.3), (5.5), and (5.6) hold, lim ˜ n nk ka →∞ = α for each k ∈N (5.10) and lim ˜ n k nka →∞ ∑ = α . (5.11) Proof. Suppose that A satisfies conditions (5.3), (5.5), (5.6), (5.10), and (5.11). Let us take any x xk= ( ) in ec r such that x lk → as k →∞ . Then Ax exists and it is trivial that the sequence y yk= ( ) connected with the sequence x xk= ( ) by relation (2.2) is in c such that y lk → as k →∞ . At this stage, we observe from (5.10) and (5.6) that j k j n j nja = ∈ ∑ ∑≤ < ∞ 0 α sup ˜ N holds for every k ∈N . This leads us to the consequence that αk( ) ∈�1. Considering (5.8), let us write k nk k k nk k k nka x a y l l a∑ ∑ ∑= −( ) +˜ ˜ , n ∈N . (5.12) In this situation, by letting n →∞ in (5.12), we observe that the first term on the right-hand side tends to k k ky l∑ −( )α by (5.6) and (5.10), and the second term tends to l α by (5.11). Now, under the light of these facts, we obtain from (5.12) as n →∞ that Ax y l ln k k k( ) → −( ) +∑α α (5.13) and this shows that A e cc r∈( ): . Conversely, suppose that A e cc r∈( ): . Then, since the inclusion c ⊂ ∞� holds, the necessities of (5.3), (5.5) and (5.6) are immediately obtained from the part (ii) of Theorem 5.1. To prove the necessity of (5.10), consider the sequence x = b rk( )( ) = = b rn k n ( )( ){ } ∈N in ec r , defined by (3.1), for every fixed k ∈N . Since Ax exists and is in c for every x ec r∈ , one can easily see that Ab r a ck nk n ( )( ) ˜= { } ∈∈N for each k ∈N which shows the necessity of (5.10). Similarly, by putting x = e in (5.8), we also obtain that Ax a k nk n = { }∑ ∈ ˜ N which belongs to the space c and this shows the necessity of (5.11). This step concludes the proof. ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 15 Let us define the concept of s-multiplicativity of a limitation matrix. When there is some notion of limit or sum in the sequence spaces λ and µ, we shall say that the method A ∈( )λ µ: is multiplicative s if every x ∈λ is A-summable to s times of lim x, for any fixed real number s and denote the class of all s-multiplicative matrices by λ µ:( )s . It is of course that the class e cc r s :( ) of s-multiplicative matrices reduces to the classes e cc r : 0( ) and e cc r reg :( ) in the cases s = 0 and s = 1, respectively; here, e cc r reg :( ) denotes the class of all matrix mappings A from ec r to c such that A x x− =lim lim for all x ec r∈ . Now, we may give the corollary to Theorem 5.2 without proof. Corollary 5.1. A e cc r s ∈( ): if and only if (5.3), (5.5), (5.6) hold, (5.10) and (5.11) also hold with αk = 0 for each k ∈N and α = s, respectively. The Steinhaus-type theorems were formulated by Maddox [17] as follows: Consider the class λ µ:( )1 of 1-multiplicative matrices and ν be a sequence space such that ν λ⊃ . Then the result of the form λ µ ν µ: :( ) ( ) = ∅1 ∩ is called a theorem of Steinhaus type, where ∅ denotes the empty set. Now, we may give a Steinhaus-type theorem whose proof requires the following lemma: Lemma 5.3 ([12], Corollary 2.5 (iii)). A e cr∈( )∞ : i f and only if (5.6), (5.10) hold, and lim ˜ lim ˜ n k nk k n nka a →∞ →∞ ∑ ∑= , (5.14) lim ( ) ˜ m k j k m j k j nj k nk j k r r a a →∞ = − −∑ ∑ ∑    − =1 , n ∈N . (5.15) Theorem 5.3. There is no matrix belonging to the classes both e cc r s :( ) and e cr ∞( ): . Proof. Suppose that the classes e cc r s :( ) and e cr ∞( ): are not disjoint. Then there is at least matrix A satisfying the conditions of both Lemma 5.3 and Corollary 5.1. Combining condition (5.10) with (5.14), one can easily see that lim ˜ n k nka →∞ ∑ = 0 which contradicts condition (5.11). This completes the proof. We now may present our basis lemma which is useful for obtaining the characterization of some new matrix classes from Theorems 5.1, 5.2 and Corollary 5.1. Lemma 5.4 ([12], Lemma 2.6). Let λ, µ be any two sequence spaces, A be an infinite matrix and B a triangle matrix. Then A B∈( )λ µ: if and only if BA ∈( )λ µ: . It is trivial that Lemma 5.4 has several consequences, some of them give the necessary and sufficient conditions of matrix mappings between the Euler sequence spaces. Indeed, combining Lemma 5.4 with Theorems 5.1, 5.2 and Corollary 5.1, one can easily derive the following results: Corollary 5.2. Let A ank= ( ) be an infinite matrix and define the matrix C cnk= ( ) by c n j t t ank j n n j j jk=     − = −∑ 0 1( ) , 0 < t < 1 and k n, ∈N . ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 16 B. ALTAY, F. BAS, AR Then the necessary and sufficient conditions in order for A belongs to anyone of the classes e ec r t: ∞( ) , e ec r p t:( ) , e ec r c t:( ) and e ec r c t s :( ) are obtained from the respective ones in Theorems 5.1, 5.2 and Corollary 5.1 by replacing the entries of the matrix A by those of the matrix C. Corollary 5.3. Let A ank= ( ) be an infinite matrix and t tk= ( ) be a sequence of positive numbers and define the matrix C cnk= ( ) by c T t ank n j n j jk= = ∑1 0 , k n, ∈N , where T tn k n k= =∑ 0 for all n ∈N . Then the necessary and sufficient conditions in order for A belongs to anyone of the classes e rc r t: ∞( ), e rc r p t:( ), e rc r c t:( ) and e rc r c t s :( ) are obtained from the respective ones in Theorems 5.1, 5.2 and Corollary 5.1 by replacing the entries of the matrix A by those of the mat- rix C ; here, rp t is defined in [18] as the space of all sequences whose Rt - transforms are in the space �p and is derived from the paranormed spaces r pt ( ) in the case p pk = for all k ∈N , and rt ∞ , rc t are obtained in the case p = e from the paranormed spaces r pt ∞( ), r pc t ( ) and are studied by Malkowsky [4]. Since the spaces rt ∞ and rp t reduce in the case t = e to the Cesàro sequence spaces X∞ and Xp of nonabsolute type, respectively, Corollary 5.3 also includes the characterizations of the classes e Xc r : ∞( ) and e Xc r p:( ). Corollary 5.4. Let A ank= ( ) be an infinite matrix and define the matrices C cnk= ( ) and D dnk= ( ) by c a ank nk n k= − +1, and dnk = ank – an k−1, for all k n, ∈N . Then the necessary and sufficient conditions in order for A belongs to anyone of the classes ec r : ( )�∞( )∆ , e cc r : ( )∆( ), e cc r s : ( )∆( ) and e bc r p: v( ) are obtained from the respective ones in Theorem 5.2, Corollary 5.1 and Theorem 5.1 by replacing the entries of the matrix A by those of the matrices C and D ; here, �∞( )∆ , c( )∆ denote the difference spaces of all bounded, convergent sequences and introduced by Kızmaz [19]. Corollary 5.5. Let A ank= ( ) be an infinite matrix and define the matrix C cnk= ( ) by c t n ank j n j jk= + += ∑ 0 1 1 , 0 < t < 1, for all k n, ∈N . Then the necessary and sufficient conditions in order for A belongs to anyone of the classes e ac r t: ∞( ) , e ac r p t:( ) , e ac r c t:( ) and e ac r c t s :( ) are obtained from the respective ones in Theorems 5.1, 5.2 and Corollary 5.1 b y replacing the entries of the matrix A by those of the matrix C. Corollary 5.6. Let A ank= ( ) be an infinite matrix and define the matrix C cnk= ( ) by c a n knk = ( ), for all k n, ∈N . Then the necessary and sufficient conditions in order for A belongs to anyone on the classes e bsc r :( ), e csc r :( ) and e csc r s :( ) are obtained from the respective ones in Theorems 5.1, 5.2 and Corol- lary 5.1 by replacing the entries of the matrix A by those of the matrix C. ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 SOME EULER SEQUENCE SPACES OF NONABSOLUTE TYPE 17 1. Choudhary B., Nanda S. Functional analysis with applications. – New Delhi: John Wiley & Sons Inc., 1989. 2. Wang C.-S. On Nölund sequence spaces // Tamkang J. Math. – 1978. – 9. – P. 269 – 274. 3. Ng P.-N., Lee P.-Y. Cesàro sequences spaces of non-absolute type // Comment. Math. Prace Mat. – 1978. – 20, # 2. – P. 429 – 433. 4. Malkowsky E. Recent results in the theory of matrix transformations in sequence space // Mat. Vesnik. – 1997. – 49. – P. 187 – 196. 5. Bas, ar F., Altay B. On the space of sequences of p-bounded variation and related matrix mappings // Ukr. Math. J. – 2003. – 55, # 1. – P. 136 – 147. 6. Aydın C., Bas, ar F. On the new sequence spaces which include the spaces c0 and c // Hokkaido Math. J. – 2004. – 33, # 1. – P. 1 – 16. 7. Aydın C., Bas, ar F. Some new sequence spaces which include the spaces �p and �∞ (under communication). 8. Aydın C., Bas, ar F. Some new difference sequence spaces // Appl. Math. Comput. – 2004 (to appear). 9. Aydın C., Bas, ar F. Some new paranormed sequence spaces // Inform. Sci. – 2004. – 160. – P. 27 – 40. 10. Altay B., Bas, ar F., Mursaleen. On the Euler sequence spaces which include the spaces �p and �∞ . I (under communication). 11. Powell R. E., Shah S. M. Summability theory and its applications. – London: Van Nostrand Reinhold Company, 1972. 12. Bas, ar F., Altay B. On the Euler sequence spaces which include the spaces �p and �∞ . II (under communication). 13. Maddox I. J. Elements of functional analysis. – 2-nd ed. – Cambridge: Univ. Press, 1988. 14. Wilansky A. Summability through functional analysis // North-Holland Math. Stud. – 1984. – 85. 15. Kamthan P. K., Gupta M. Sequence spaces and series. – New York; Basel: Marcel Dekker Inc., 1981. 16. Stieglitz M., Tietz H. Matrixtransformationen von Folgenräumen Eine Ergebnisübersict // Math. Z. – 1977. – 154. – S. 1 – 16. 17. Maddox I. J. On theorems of Steinhaus type // J. London Math. Soc. – 1967. – 42. – P. 239 – 244. 18. Altay B., Bas, ar F. On the paranormed Riesz sequence spaces of non-absolute type // Sotutheast Asian Bull. Math. – 2002. – 26, # 5. – P. 701 – 715. 19. Kızmaz H. On certain sequence spaces // Can. Math. Bull. – 1981. – 24, # 2. – P. 169 – 176. Received 11.09.2003 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1

On Some Euler Sequence Spaces of Nonabsolute Type

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