ON THE SPACES OF EULER ALMOST NULL AND EULER ALMOST CONVERGENT SEQUENCES*

Let Er denotes the Euler means of order r. The Euler sequence spaces e0, e r c and e r p, e r ∞ consisting of all sequences whose E r-transforms are in the spaces c0, c and `p, `∞ are introduced by Altay and Başar [2], Altay et al. [3], and Mursaleen et al. [22]. Recently, Polat and Başar have studied the Euler spaces of difference sequences of order m, in [24]. The concept almost convergence of a bounded sequence introduced by Lorentz [19]. Quite recently, Başar and Kirişci have worked the domain of the generalized difference matrix B(r, s) in the sequence spaces f0 and f of almost null and almost convergent sequences, in [8]. In this paper, following Başar and Kirişci [8], we essentially deal with the domains (f0)Er and fEr of the Euler means of order r in the spaces f0 and f . Therefore, we add two new spaces to the Euler sequence spaces.


Introduction
By a sequence space, we understand a linear subspace of the space != C N of all complex sequences which contains , the set of all …nitely non-zero sequences, where C denotes the complex …eld and N = f0; 1; 2; : : :g.We write `1, c and c 0 for the classical spaces of all bounded, convergent and null sequences, respectively.Also by bs, cs, `1 and `p, we denote the space of all bounded, convergent, absolutely and p-absolutely convergent series, respectively.
Let and be two sequence spaces, and A = (a nk ) be an in…nite matrix of complex numbers a nk , where k; n 2 N.Then, we say that A de…nes a matrix mapping from into , and we denote it by writing A : !if for every sequence x = (x k ) 2 .The sequence Ax = f(Ax) n g, the A-transform of x, is in ; where For simplicity in notation, here and in what follows, the summation without limits runs from 0 to 1.By ( : ), we denote the class of all matrices A such that A : ! .Thus, A 2 ( : ) if and only if the series on the right side of (1.1) converges for each n 2 N and each x 2 and we have Ax = f(Ax) n g n2N 2 for all x 2 .A sequence x is said to be A-summable to l if Ax converges to l which is called the A-limit of x.If there is some notion of limit or sum in and , then we write ( ; ; p) to denote the subclass of ( : ), which preserves the limit or sum.Further, A 2 ( : c) is said to be strongly-multiplicative s, if lim Ax = s(f lim x k ) for each x = (x k ) 2 , where 2 ff; f (E)g.By ( : ) s , we denote the class of all such matrices.It is now trivial in the case s = 1 that the class ( ; ) s coincides with the class ( ; ; p) and thus it is immediate that ( ; ; p) ( ; ) s ( ; ).The matrix domain A of an in…nite matrix A in a sequence space is de…ned by which is a sequence space.If A = (a nk ) is triangle, i.e., a nn 6 = 0 and a nk = 0 for all k > n, then one can easily observe that the sequence spaces A and are linearly isomorphic, i.e., A = .
The main purpose of present paper is to introduce the spaces f 0 (E) and f (E) of Euler almost null and Euler almost convergent sequences, and to determine the -and -duals of these spaces.Furthermore, some classes of matrix mappings on the space of Euler almost convergent sequences are characterized.
We shall write throughout for brevity that for all k; m; n 2 N.

Euler Sequence Spaces
Firstly, we give the de…nitions of some sequence spaces in the existing literature.The Euler sequence spaces e r 0 and e r c were de…ned by Altay and Başar [2] and the spaces e r p and e r 1 were de…ned by Altay et al. [3], as follows: ) ; where E r = (e r nk ) denotes the Euler means of order r de…ned by for all k; n 2 N. It is known that the method E r is regular for 0 < r < 1 and E r is invertible such that (E r ) 1 = E 1=r with r 6 = 0. We assume unless stated otherwise that 0 < r < 1.
Altay and Başar [2] gave the inclusion relations between the sequence spaces e r 0 and e r c with the classical sequence spaces, determined the Schauder basis for these spaces.They also calculated the alpha-, beta-, gamma-and continuous duals of the Euler sequence spaces, and characterized some matrix mappings on e r 0 and e r c .Altay et al. [3] calculated the dual spaces of the sequence spaces e r p and e r 1 , and constructed the Schauder basis of the sequence space e r p .In [22], Mursaleen et al. characterized the classes (e r p : `1), (e r 1 : `p) and (e r p : f ) of in…nite matrices for 1 < p 1 and gave the characterizations of some other matrix mappings from the space e r p to the Euler, Riesz, di¤erence, etc., sequence spaces, also Mursaleen et al. [22] emphasized on some geometric properties such as Banach-Saks property, weak Banach-Saks property, …xed point property, Banach-Saks type p of the space e r p .Kara et al. [15] introduced the Euler sequence spaces e r (p) of nonabsolute type and proved that the spaces e r (p) and `(p) are linearly isomorphic.Also the alphabeta-and gamma-duals of the Euler sequence spaces e r (p) of nonabsolute type are computed in [15].Kara et al. [15] de…ned a modular on the generalized Euler sequence spaces e r (p) and considered it equipped with the Luxemburg norm.Therefore, they gave some relationships between the modular and Luxemburg norm on the space e r (p) has property (H) but is not rotund (R).
Let m be a positive integer.We de…ne the operators (m) ; P (m) : !! ! b (1) x x j for all k 2 N; (m) x = (1) ( (m 1) )x; (m) X (m 1) X 1 A x for all m 2: The following equalities hold for m 1 and k = 0; 1; 2; : : : where I is the identity on !.We write and for the matrices with nk = (1) (e (k)  n and ) n for all n; k 2 N.So the operators (1) and P (1) are given by the matrices and P .Similarly, the operators (m) and are given by the composition of and P with themselves m times.
Altay and Polat [4] de…ned the Euler sequence spaces with di¤erence operator as follows: ; where x k = x k x k 1 .Following Altay and Polat [4], Polat and Başar [24] gave the new sequence spaces e r 0 ( (m) ), e r c ( (m) ) and e r 1 ( (m) ) consisting of all sequences x = (x k ) such that their (m) transforms are in Euler the spaces e r 0 , e r c and e r 1 , respectively, that is, The sequence spaces e r 0 ( (m) ), e r c ( (m) ) and e r 1 ( (m) ) are reduced in the case m = 1 to the spaces e r 0 ( ), e r c ( ) and e r 1 ( ) of Altay and Polat [4].
Başar¬r and Kay¬kç¬ [10] de…ned the matrix for all k; n 2 N which is reduced to the mth order di¤erence matrix (m) in case r = 1, s = 1, where (m) = ( (m 1) ) and m 2 N. Kara and Başar¬r [16] introduced the B m Euler di¤erence sequence spaces e r 0 (B (m) ), e r c (B (m) ) and e r 1 (B (m) ) as the set of all sequences whose B m transforms are in the Euler spaces e r 0 , e r c and e r 1 , respectively, that is, Karakaya and Polat [17] de…ned the new paranormed Euler sequence spaces with di¤erence operator as follows: ) The new sequence spaces e r 0 ( ; p), e r c ( ; p) and e r 1 ( ; p) are reduced to some sequence spaces corresponding to special cases of (p k ).For instance, in the case p k = 1 for all k 2 N, the sequence spaces e r 0 ( ; p), e r c ( ; p) and e r 1 ( ; p) are reduced to the sequence spaces e r 0 ( ), e r c ( ) and e r 1 ( ) de…ned by Altay and Polat [4].Demiriz and Çakan [11] introduced the sequence spaces e r 0 (u; p) and e r c (u; p) of nonabsolute type, as the sets of all sequences such that their E r;u transforms are in the spaces c 0 (p) and c(p), respectively, that is, ) where u = (u k ) is the sequence of non-zero reals.In the case (u k ) = (p k ) = e = (1; 1; 1; : : :), the sequence spaces e r 0 (u; p) and e r c (u; p) are, respectively, reduced to the sequence spaces e r 0 and e r c introduced by Altay and Başar [2].Djolović and Malkowsky [12] added a new supplementary aspect to research of Polat and Başar [24] by characterizing classes of compact operators on those spaces.In [12], the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 ; c and `1.The main tool for their characterizations is the Hausdor¤ measure of noncompactness.

Spaces of Euler Almost Null and Euler Almost Convergent Sequences
In this section, we study some properties of the spaces of the almost null and almost convergent Euler sequences.
The shift operator P is de…ned on ! by (P x) n = x n+1 for all n 2 N. A Banach limit L is de…ned on `1 as a non-negative linear functional, such that L(P x) = L(x) and L(e) = 1.A sequence x = (x k ) 2 `1 is said to be almost convergent to the generalized limit l if all Banach limits of x is l [19], and is denoted by f lim x k = l.Let P i be the composition of P with itself i times and write for a sequence x = (x k ) Lorentz [19] proved that f lim x k = l if and only if lim m!1 t mn (x) = l uniformly in n.It is well-known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal.By f and f s, we denote the space of all almost convergent sequences and series, respectively, i.e., It is proved in [8] that f is a Banach space with the norm where t mn (x) is de…ned as in (3.1).Başar and Kirişci [8] have de…ned the sequence spaces b f 0 and b f derived by the domain of generalized di¤erence matrix B(r; s) in the sequence spaces f 0 and f , that is b where the generalized di¤erence matrix B(r; s) = fb nk (r; s)g is de…ned by for all k; n 2 N. We introduce the sequence spaces f 0 (E) and f (E) as the sets of all sequences whose E r -transforms are in the spaces f 0 and f , that is With the notation of (1.2), we can rede…ne the spaces f 0 (E) and f (E) as follows: De…ne the sequence y = fy k (r)g by the E r transform of a sequence x = (x k ), i.e., (1 r) k j r j x j for all k 2 N: is a norm on the spaces f 0 (E) and f (E), where kxk f (E) = sup m;n2N jt mn (y)j.Now we give some inclusion relations between the sequence spaces f 0 (E), f (E), c and `1.
Proof.It is clear that f (E) `1.Now, we should show that this inclusion is strict.De…ne the sequence x = E 1=r y with the sequence y in the set `1 n f given by Miller and Orhan [21] as y = f0; : : : ; 0; 1; : : : ; 1; 0; : : : ; 0; 1; : : : ; 1; : : :g, where the blocks of 0's are increasing by factors of 100 and blocks of 1's are increasing by factors of 10.Then, the sequence x is not in f (E) but in the space `1, as desired.
Proof.It is clear that c f (E).Now we show that this inclusion is strict.Now we consider the sequence x = (x k ) de…ned by x k (r) = ( r) k for all k 2 N. The sequence is not convergent but is in the space f (E).
Theorem 3.3.The spaces f 0 (E) and f (E) are linearly isomorphic to the spaces f 0 and f , respectively, i.e., f 0 (E) = f 0 and f (E) = f .Proof.To prove this theorem, we should show the existence of a linear bijection between the spaces f (E) and f .Consider the transformation T from f (E) to f by y = T x = E r x.The linearity of T is clear.Further, it is obvious that x = whenever T x = and hence T is injective.
Let us take any y 2 f and de…ne the sequence x = fx k (r)g by 1) k j r k y j for all k 2 N: Then, one can see that which shows that E r x 2 f , i.e., x 2 f (E).Consequently, we see from here that T is surjective.Hence T is a linear bijection which therefore says us that the spaces f (E) and f are linearly isomorphic, as was desired.
Since one can show in the similar way that f 0 (E) = f 0 , we omit the detail.
Başar and Kirişci [8] proved that sequence space f is a BK space with the norm k k 1 and non-separable closed subspace of (`1; k k 1 ).So, the sequence space f has no Schauder basis.Jarrah and Malkowsky [1] showed that the matrix domain A of a normed sequence space has a basis if and only if has a basis whenever A = (a nk ) is triangle.Then; The sequence spaces f 0 (E) and f (E) have no Schauder basis.

Duals of the Spaces of Euler Almost Null and Euler Almost Convergent Sequences
The set S( ; ) de…ned by is called the multiplier space of the sequence spaces and .One can eaisly observe for a sequence space with that the inclusions S( ; ) S( ; ) and S( ; ) S( ; ) hold.With the notation of (4.1), the alpha-, beta-and gamma-duals of a sequence space , which are respectively denoted by , and are de…ned by = S( ; `1); = S( ; cs) and = S( ; bs): The alpha-, beta-and gamma-duals of a sequence space are also referred as Köthe-Toeplitz dual, generalized Köthe-Toeplitz dual and Garling dual of a sequence space, respectively.We give the beta-and gamma-duals of the sequence spaces f 0 (E) and f (E).For this, we need the following lemma: Lemma 4.1.Let A = (a nk ) be an in…nite matrix.Then, the following statements hold: (ii) (cf.[25]).A 2 (f : c) if and only if (4.2) holds and (iii) (cf.[13]).A 2 (f : f ) if and only if (4.2) holds and (iv) (cf.[13]).A 2 (`1 : f ) if and only if (4.2), (4.6) and (4.8) hold.; ; Then, the dual of the sequence space f (E) is Proof.Let a = (a k ) 2 ! and consider the equality where T r = (t r nk ) is de…ned by for all k; n 2 N. Thus, we deduce from Part (ii) of Lemma 4.1 with (4.9) that ax = (a k x k ) 2 cs whenever x = (x k ) 2 f (E) if and only if T r y = f(T r y) n g 2 c whenever y = (y k ) 2 f , where T r = (t r nk ) is de…ned by (4.10).Therefore, we derive from (4.2), (4.3), (4.4) and ( 4 Proof.This is similar to the proof of Theorem 4.2 with Part (i) of Lemma 4.1 instead of Part (ii) of Lemma 4.1.So, we omit the detail.

Matrix transformations Related to the Sequence Space f (E)
In the present section, we characterize the matrix transformations from f (E) into any given sequence space .
Since f (E) = f , it is trivial that the equivalence "x 2 f (E) if and only if y 2 f " holds.Proof.Let be any given sequence space.Suppose that (5.1) holds between A = (a nk ) and D = (d nk ), and take into account that the spaces f (E) and f are linearly isomorphic.
Let A 2 (f (E) : ) and take any y = (y k ) 2 f .Then DE r exists and fa nk g k2N 2 T 5 i=1 d r i which yields that fd nk g k2N 2 `1 for each n 2 N. Hence, Dy exists and thus for all n 2 N. We have that Dy = Ax which leads us to the consequence D 2 (f : ).Conversely, let fa nk g k2N 2 ff (E)g for each n 2 N and D 2 (f : ) hold, and take any x = (x k ) 2 f (E).Then, Ax exists.Therefore, we obtain from the equality Let be any given sequence space.Then, A = (a nk ) 2 ( : Proof.Let z = (z k ) 2 and consider the following equality which yields as m ! 1 that (Bz) n = fE r (Az)g n for all n 2 N. Therefore, one can observe from here that Az 2 f (E) whenever z 2 if and only if Bz 2 f whenever z 2 .This completes the proof.
Of course, Theorems 5.1 and 5.2 have several consequences depending on the choice of the sequence space .Whence by Theorem 5.1 and Theorem 5.2, the necessary and su¢ cient conditions for (f (E) : ) and ( : f (E)) may be derived by replacing the entries of C and A by those of the entries of D = CE 1=r and B = E r A, respectively; where the necessary and su¢ cient conditions on the matrices D and B are read from the concerning results in the existing literature.Now, we list the following conditions on an in…nite matrix A = (a nk ) transforming the sequences from/in the sequence space f : (5.1) f lim a(n; k) = k exists for each …xed k 2 N ; (5.7)  Proof.Suppose that the converse of this is true, that is (f (E) : c) s T (`1 : c) 6 = ;.Then there exists at least one in…nite matrix A satisfying the conditions of Corollary 5.6 and Schur's theorem.Then, we can easily see that lim n!1 a nk = 0 which contradicts the condition lim n!1 P k a nk = s of Corollary 5.4.This completes the proof.
Proof.This is similar to the proof of Theorem 5.8.So, we omit the detail.

Conclusion
The construction of new sequence spaces with the Euler mean were studied by Altay and Başar [2], Altay et al. [3] and Mursaleen et al. [22].After Altay and Polat [4], Polat and Başar [24] studied the Euler di¤erence sequence spaces of order m.Also, Karakaya and Polat [17] extended the Euler sequence spaces e r 0 ( ), e r c ( ) and e r 1 ( ) de…ned by Altay and Polat [4] to the paranormed case.Kara et al. [15] studied some topological and geometrical properties of the generalized Euler spaces.Further Başar¬r and Kay¬kç¬ [10] de…ned Euler B (m) -di¤erence sequence spaces.Demiriz and Çakan [11] introduced the sequence spaces e r 0 (u; p) and e r c (u; p) of nonabsolute type, as the sets of all sequences such that E r;u -transforms of them are in the spaces c 0 (p) and c(p).Djolović and Malkowsky [12] added a new supplementary aspect to research of Polat and Başar [24] by characterizing classes of compact operators on those spaces.
The concept of almost convergence has been employed many mathematicians since 1948.Başar and Kirişci [8] established new almost convergent sequence spaces with the generalized di¤erence matrix B(r; s) and Sönmez [26] studied the concept of almost convergence with the triple band matrix B(r; s; t).Başar and Kirişci [8] proved that the space f is a BK space with the sup-norm, and is a non-seperable closed subspace of (`1; k k 1 ).Since the space f is non-seperable, this space and the spaces isomorphic to the space f have no Schauder basis.
In this paper, we combine the almost convergence with the Euler means.Since the domain of generalized di¤erence matrix B(r; s) in the space f is studied by Başar and Kirişci [8], the present paper is its natural continuation.
Finally, we should note from now on that the investigation of the domain of some particular limitation matrices, namely the composition of Euler means with the m th order di¤erence matrix or generalized weighted mean, the matrix , etc., in the space f will lead us to new results.Also it can study various matrix transformations, such as sequence-to-sequence, sequence-to-series, series-to-sequences and series-toseries, between the new almost Euler sequence spaces and other spaces.

Theorem 4 . 2 .
De…ne the sets d r 1 , d r 2 , d r 3 , d r 4 , d r 5 de…ned as follows: j k r j a j exists 9 =

1 ) j k r j a j k 3 5
= 0 which shows that ff (E)g = T 5 n=1 d r n .Theorem 4.3.The dual of the sequence spaces f 0 (E) and f (E) is the set d r 1 .

Theorem 5 . 1 .
Suppose that the entries of the in…nite matrices A = (a nk ) and D = (d nk ) are connected with the relationd nk = a nkfor all k; n 2 N and be any given sequence space.Then A 2 (f (E) : ) if and only if fa nk g k2N 2 f (E) for all n 2 N and D 2 (f : ).

3 5 y k for all n 2 NTheorem 5 . 2 .
, as m ! 1 that Dy = Ax and this shows that A 2 (f (E) : ).This completes the proof.By changing the roles of the spaces f (E) with in Theorem 5.1, we have: Suppose that the elements of the in…nite matrices A = (a nk ) and B = (b nk ) are connected with the relation b nk := n X j=0 n j(1 r) n j r j a jk for all k; n 2 N:

Corollary 3 .
A 2 (f (E) : c) s if and only if (4.2) holds, (4.3) and (4.5) hold with k = 0 for each k 2 N and (4.4) also holds with = s with a nk instead of a nk .Corollary 4. A 2 (f (E) : f ) s if and only if (4.2), (4.6) and (4.8) hold with k = 0 and (4.7) also holds with = s with a nk instead of a nk .Now, we may mention about Steinhaus type theorems which were formulated by Maddox [20], as follows: Consider the class ( : ) 1 of 1-multiplicative matrices and be a sequence space such that .Then a result of the form ( : ) 1 T ( : ) = ;, where ; denotes the empty set, is called a theorem of the Steinhaus type.Now, we can give the next Steinhaus type theorem concerning with the stronglymultiplicative and coercive matrix classes: Theorem 5.4.The classes (f (E) : c) s and (`1 : c) are disjoint.
Corollary 1.Let A = (a nk ) be an in…nite matrix.The following statements hold:(i) A 2 (f (E) : `1)if and only if fa nk g k2N 2 ff (E)g for all n 2 N and (4.2) holds with a nk instead of a nk .(ii) A 2 (f (E) : c) if and only if fa nk g k2N 2 ff (E)g for all n 2 N and (4.2)-(4.5)hold with a nk instead of a nk .(iii) A 2 (f (E) : c 0 ) if and only if fa nk g k2N 2 ff (E)g for all n 2 N and (4.2) holds, (4.3) and (4.5) hold with k = 0, and (4.4) holds and = 0 as a nk instead of a nk .(iv) A 2 (f (E) : f ) if and only if fa nk g k2N 2 ff (E)g for all n 2 N and (4.2), (4.6)-(4.8)hold with a nk instead of a nk .(v) A 2 (f (E) : bs) if and only if fa nk g k2N 2 ff (E)g for all n 2 N and (5.10) holds.(vi) A 2 (f (E) : cs) if and only if fa nk g k2N 2 ff (E)g for all n 2 N and (5.10)-(5.13)hold with a nk instead of a nk .Corollary 2. Let A = (a nk ) be an in…nite matrix and b nk be de…ned by (5.2). nk instead of a nk .(ii) A = (a nk ) 2 (f : f (E)) if and only if (4.2), (4.6), (4.7) and (4.8) hold with b nk instead of a nk .(iii) A = (a nk ) 2 (c : f (E)) if and only if (4.2), (4.6) and (4.7) hold with b nk instead of a nk .(iv) A = (a nk ) 2 (bs : f (E)) if and only if (5.2), (5.3), (4.6) and (5.6) hold with b nk instead of a nk .(v) A = (a nk ) 2 (f s : f (E)) if and only if (5.3), (4.6), (4.8) and (5.6) hold with b nk instead of a nk .(vi) A = (a nk ) 2 (cs : f (E)) if and only if (5.2) and (4.6) hold with b nk instead of a nk .(vii) A = (a nk ) 2 (bs : f s(E)) if and only if (5.3), (5.6)-(5.8)hold with b nk instead of a nk , where f s(E) denotes the domain of the matrix E r in the sequence space f s. (viii) A = (a nk ) 2 (f s : f s(E)) if and only if (5.6)-(5.9)hold with b nk instead of a nk .(ix) A = (a nk ) 2 (cs : f s(E)) if and only if (5.7) and (5.8) hold with b nk instead of a nk .Now, we can give some consequences, below: Then, following statements hold:(i) A = (a nk ) 2 (`1 : f (E))if and only if (4.2), (4.6) and (5.5) hold with b