Research Article On the Riesz Almost Convergent Sequences Space

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1 Abstract and Applied Analysis Volume 202, Article ID 69694, 8 pages doi:0.55/202/69694 Research Article On the Riesz Almost Convergent Sequences Space Mehmet Şengönül and Kuddusi Kayaduman 2 Faculty of Science and Letters, Nevşehir University, Nevşehir, Turey 2 Faculty of Arts and Sciences, University of Gaziantep, 2730 Gaziantep, Turey Correspondence should be addressed to Mehmet Şengönül, msengonul@yahoo.com Received 8 January 202; Revised March 202; Accepted 4 March 202 Academic Editor: Norimichi Hirano Copyright q 202 M. Şengönül and K. Kayaduman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. The purpose of this paper is to introduce new spaces f and f 0 that consist of all sequences whose Riesz transforms of order one are in the spaces f and f 0, respectively. We also show that f and f 0 are linearly isomorphic to the spaces f and f 0, respectively. The β-andγ-duals of the spaces f and f 0 are computed. Furthermore, the classes f : μ and μ : f of infinite matrices are characterized for any given sequence space μ and determine the necessary and sufficient conditions on a matrix A to satisfy B R core Ax K core x, B R core Ax st core x for all x l.. Introduction and Preinaries Let w be the space of all real or complex valued sequences. Then, each linear subspace of w is called a sequence space. For example, the notations l, c, c 0, l, cs, andbs are used for the sequence spaces of all bounded, convergent, and null sequences, absolutely convergent series, convergent series, and bounded series, respectively. Let λ and μ be two sequence spaces and A a n an infinite matrix of real or complex numbers a n, where n, N {0,, 2,...}. Then, A defines a matrix mapping from λ to μ andisdenotedbya : λ μ if for every sequence x x λ the sequence Ax { Ax n },thea-transform of x,isinμ where Ax n a n x, n N.. By λ : μ, we denote the class of matrices A such that A : λ μ. Thus,A λ : μ if and only if the series on the right side of. converges for each n N and every x λ, and we have Ax { Ax n } n N μ for all x λ. Thematrix domain λ A of an infinite matrix A in a sequence space λ is defined by λ A {x x w : Ax λ}..2

2 2 Abstract and Applied Analysis If we tae λ c, then c A is called, convergence domain of A, and we write the it of Ax as A x a nx. Further A is called regular if A x x for each convergent sequence x. Let λ be a sequence space. Then λ is called solid if and only if l λ λ,. The approach constructing a new sequence space by means of the matrix domain of a particular itation method has recently been employed by Altay and Başar 2,Başarir 3,Aydınand Başar 4, Kirişçi and Başar 5,Şengönül and Başar 6, Polat and Başar 7, and Malowsy et al. 8. Finally, the new technique for deducing certain topological properties, such as AB-, KB-, and AD-properties, and solidity and monotonicity, and determining the α-, β- and γ- duals of the domain of a triangle matrix in a sequence space is given by Altay and Başar 9. Furthermore, quite recently, Kirişçi and Başar 0 introduced the new sequence space f derived from the space f of almost convergent sequences by means of the domain of the generalized difference matrix B r, s. Define the sets f and f 0 by { f x x w : α C { f 0 x x w : 0 0 } x p α uniformly in p, n x p 0 uniformly in p n }..3 If x f, then x is said to be almost convergent to the generalized it α. When x f, we write f x α. Lorentz introduced this concept and obtained the necessary and sufficient conditions for an infinite matrix to contain f in its convergence domain. These conditions on an infinite matrix A a n consist of the standard Silverman Toeplitz conditions for regularity plus the condition a n a n, 0. Such matrices are called strongly regular.one of the best nown strongly regular matrices is C, thecesàro matrix of order one which is a lower triangular matrix defined by c n n, 0 n, 0, > n,.4 for all n, N. A matrix U is called the generalized Cesàro matrix if it is obtained from C by shifting rows. Let p : N N. Then, U u n is defined by, p n p n n, u n n 0, otherwise,.5 for all n, N.

3 Abstract and Applied Analysis 3 Let us pose that G is the set of all such matrices obtained by using all possible functions p. Now, right here, let us give a new definition for the set of almost convergent sequences that was introduced by Butović etal. 2 : Lemma.. The set f of all almost convergent sequences is equal to the set U G c U. Other one of the best nown regular matrices is R r n, the Riesz matrix which is a lower triangular matrix defined by r, 0 n, r n R n 0, > n,.6 for all n, N, where r is real sequence with r 0 N and R n r 0 r r n. Let K be a subset of N. The natural density δ of K is defined by δ K { n : K}, n.7 where the vertical bars indicate the number of elements in the enclosed set. The sequence x x is said to be statistically convergent to the number l if, for every ε, δ { : x l ε} 0 see 3. In this case, we write st x l. We will also write S and S 0 to denote the sets of all statistically convergent sequences and statistically null sequences. The statistically convergent sequences were studied by several authors see 3, 4 and others. Let us consider the following functionals defined on l : l x inf x, L x x, p q σ x p n N p n N x σ i n, L p x x n i. p p.8 In 5, the σ-core of a real bounded sequence x is defined as the closed interval q σ x,q σ x and also the inequalities q σ Ax L x σ-core of Ax K-core of x q σ Ax q σ x σ-core of Ax σ-core of x, for all x l, have been studied. Here the Knopp core, in short K-core, of x is the interval l x,l x. In particular, when σ n n, since q σ x L x, σ-core of x is reduced to the Banach core, in short B-core, of x defined by the interval L x,l x. The concepts of B-core and σ-core have been studied by many authors 6, 7. Recently, Fridy and Orhan 3 have introduced the notions of statistical boundedness, statistical it erior or briefly st, and statistical it inferior or briefly st inf, defined the statistical core or briefly st-core of a statistically bounded sequence as the closed interval st inf x, st x, and also determined the necessary and sufficient conditions for a matrix A to yield K-core Ax st-core x for all x l.

4 4 Abstract and Applied Analysis Let us write T x x n. p N 0.9 Quite recently, B C -core of a sequence has been introduced by the closed intervals T x, T x and also the inequalities T Ax L x,l Ax T x,t Ax T x,t Ax st x.0 have been studied for all x l in 8. Definition.2. Let x l. Then, B R -core of x is defined by the closed interval τ x,τ x, where τ x τ x inf n p N p N n p, p.. Therefore, it is easy to see that B R -core of x is l if and only if f x l. As nown, the method to obtain a new sequence space by using the convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of the convergence field of an infinite matrix in the space of almost convergent sequences is new. 2. The Sequence Spaces f and f 0 In this section we introduce the new spaces f and f 0 as the sets of all sequences such that their R-transforms are in the spaces f and f 0, respectively, that is f f 0 x x w : α C x x w : n n p α uniformly in p, p α uniformly in p. 2.

5 Abstract and Applied Analysis 5 With the notation of.2, we can write f f R and f 0 f 0 R. Define the sequence y y, which will be frequently used, as the R-transform of a sequence x x,thatis, y n 0 r x R n n N. 2.2 If R C, which is Cesàro matrix, order, then the space f and f 0 correspond to the spaces f and f 0 see 8. Suppose that G {G : G U R, U G and R is Riesz matrix}. Then we have the following proposition. Proposition 2.. f T G c T dir. Proof. The proof is similar to the proof of Lemma. so we omit the details, see 2. Consider the function f : f R, anddefine x f n n p. 2.3 The function f is a norm and f, f is BK-space. The proof of this is as follows. Theorem 2.2. The sets f and f 0 are linear spaces with the coordinate wise addition and scalar multiplication that is the BK- space with the x f Rx f. Proof. The first part of the theorem can be easily proved. We prove the second part of the theorem. Since.2 holds and f and f 0 are the BK-spaces with respect to their natural norm, also the matrix R is normal and gives the fact that the spaces f and f 0 are BK-spaces. Theorem 2.3. The sequence spaces f and f 0 are linearly isomorphic to the spaces f and f 0, respectively. Proof. Since the fact the spaces f 0 and f 0 are linearly isomorphic can also be proved in a similar way, we consider only the spaces f and f. In order to prove the fact that f f, we should show the existence of a linear bijection between the spaces f and f. Consider the transformation T defined, with the notation of 2.2,from f to f by x y Tx. The linearity of T is clear. Further, it is trivial that x θ 0, 0,... whenever Tx θ and hence T is injective. Let y y f, and define the sequence x x by x r ( R y R y ), N. 2.4

6 6 Abstract and Applied Analysis Then, we have f R x n n n p uniformly in p ( r i /ri ( )) R i y i p R i y i p uniformly in p y uniformly in p 2.5 f y uniformly in p, which shows that x f. Consequently, we see that T is surjective. Hence, T is a linear bijection that therefore shows that the spaces f and f are linearly isomorphic, as desired. This completes the proof. Theorem 2.4. The spaces f and f 0 are not solid sequence spaces. Proof. If we tae u u R 0 /r 0, R 0 /r 0 R /r, R /r R 2 /r 2,..., R /r R /r,... and v v,,,...,,..., then we see that u f and v l.letuv be t, thatis, uv t. Then, since t R 0 /r 0, R 0 /r 0 R /r, R /r R 2 /r 2,..., R /r R /r,... and it is not hard to see by taing into account the definition Riesz matrix that f R t n / n j r i t i p /. This shows that the multiplication l f of the spaces l and f is not a subset of f and therefore the space f is not solid. The proof for the space f 0 is similar to the proof of the space f,sowe omit it. Theorem 2.5. Let the spaces f and f 0 be given. Then, the inclusion f 0 f holds and the space f is not a subset of the space l, 2 if /R n c and r l,thenl f strictly holds. Proof. Clearly, the inclusion f 0 f holds. Let us consider the sequence given by u R r ( R r R ), is odd, r R r, is even, 2.6 Since R, the sequence u is not a bounded sequence. But clearly u f. This shows that to us, the space f is not a subset of the space l. 2 If /R n c and r l, then for all x l we have Rx c. Therefore, since Rx f Rx, weseethatx f.

7 Abstract and Applied Analysis 7 In Theorem 2.6, we will use some similar techniques that are due to Móricz and Rhoades 9. Theorem 2.6. Define the sequences α n and β n by α n inf β n n n p, p 2.7 for all n N. Then, α n β n for each n N and i the sequence α 2 n is nondecreasing, ii the sequence β 2 n is nonincreasing. Proof. It is trivial that α n inf n p n p β n 2.8 for each n N. Since the part ii can be proved in a similar way, we prove only part i α 2 n p 2 n inf 2 n inf inf inf 2 n 2n 2 n p 2n 2 n 2 n p 2n p 2n 2 n 2 n 2 n p 2n 2 n 2 n 2 n 2 n 2.9 This step completes the proof. 2 n 2n α 2 n 2 n α 2 n α 2 n.

8 8 Abstract and Applied Analysis Theorem 2.7. p β 2 p α 2 p 0 if and only if x f. Proof. Suppose that β 2 n α 2 n 0. For each n, choose r to satisfy 2 r n 2 r.we may write n in a dyadic representation of the form n r n i2 i, where each n i is 0 or, i 0,, 2,...,r, and n r. Then, p n n n p r n i2 i n 2 0 p n 02 0 n j n 2 2 r n r2 r j n r 2 r ( ) 2 0 ( ) 2 2 r [ ( ) ] 2 r α 2 r 2 r α 2 r 2 0 α 2 0 n r n ) n j (2 j α j 2.0 since n j {0, }, and hence p n inf n p n n α n n r ) n j (2 j α 2 j, r ( β n n j 2 j ) α n 2 j. 2. Thus, 0 β n α n n r (β2 n j (2 j ) ) j α 2 j. 2.2 If T is the lower triangular matrix with nonzero entries t n n 2 / n, then, T is a regular matrix so that r β 2 r α 2 r 0. From the equality 2.2, we see that β n α n 0. Conversely, assume that x f. Then, since p n n α 2.3

9 Abstract and Applied Analysis 9 implies α n inf p n β n p n n n α, α, 2.4 we have α n β n ( αn β n ) If we tae n 2 p, then the proof of sufficiency is obtained. This step completes the proof. 3. Some Duals of the Spaces f and f 0 In this section, by using techniques in 9, we have stated and proved the theorems determining the β- andγ-duals of the spaces f 0 and f. For the sequence spaces λ and μ, define the set S λ, μ by S ( λ, μ ) { z z w : xz x z μ x x λ }. 3. With the notation of 3., theα-, β-, and γ-duals of a sequence space λ, which are, respectively, denoted by λ α, λ β,andλ γ, are defined by λ α S λ, l, λ β S λ, cs, λ γ S λ, bs. 3.2 The following two lemmas are introduced in 20 which we need in proving Theorems 3.3 and 3.4. Lemma 3.. A f : l if and only if n N a n <. 3.3 Lemma 3.2. A f : c if and only if a n a, a n a ; N, Δ a n a

10 0 Abstract and Applied Analysis Theorem 3.3. The γ-duals of the spaces f and f 0 are the set d d 2,where { d a a w : ( ) R a Δ < }, r { { } } an d 2 a a w : R n l. r n 3.5 Proof. Define the matrix T t n via the sequence a a w by ( a ) R Δ, 0 n, r t n a n R n, n, r n 0, otherwise, 3.6 for all n, N. Here, Δ a /r a /r a /r.byusing 2.2, we derive that n a x R Δ 0 ( a ) y a n r n R n y n r ( Ty ) n, n N. From 3.7, weseethatax a x bs whenever x x f if and only if Ty l whenever y y f. Then, we derive by Lemma 3. that R Δ 0 ( a r ) <, { } an R n l, r n 3.8 which yields the desired result f γ f γ 0 d d 2. Theorem 3.4. Define the set d 3 by { d 3 a a w : [R Δ Δ ( a r ) ] } a <. 3.9 Then, f β d 3 cs. Proof. Consider equality 3.7, again. Thus, we deduce that ax a x cs whenever x x f if and only if Ty c whenever y y f. It is obvious that the columns of that matrix T, defined by 3.6, are in the space c. Therefore, we derive the consequence from Lemma 3.2 that f β d 3 cs.

11 Abstract and Applied Analysis 4. Some Matrix Mappings Related to the Spaces f and f 0 In this section, we characterize the matrix mappings from f into any given sequence space via the concept of the dual summability methods of the new type introduced by Başar 2. Note that some researchers, such as, Başar 2, Başar and Çola 22, Kuttner 23, and Lorentz and Zeller 24, wored on the dual summability methods. Now, following Başar 2, we give a short survey about dual summability methods of the new type. Let us pose that the infinite matrices A a n and B b n map the sequences x x and y y, which are connected by the relation 2.2 to the sequences z z n and t t n, respectively, that is, z n Ax n a n x, n N, t n ( By ) n b n y, n N. 4. It is clear here that the method B is applied to the R-transform of the sequence x x while the method A is directly applied to the entries of the sequence x x. So, the methods A and B are essentially different. Let us assume that the matrix product BR exists, which is a much weaer assumption than the conditions on the matrix B belonging to any matrix class, in general. The methods A and B in 4., 4.2 are called dual summability methods of the new type if z n reduces to t n or t n reduces to z n under the application of formal summation by parts. This leads us to the fact that BR exists and is equal to A and BR x B Rx formally holds if one side exists. This statement is equivalent to the following relation between the entries of the matrices A a n and B b n : a n : j ( r j an b nj or b n : R r a ) n, R Δ r ( an r ) 4.2 for all n, N. Now, we give the following theorem concerning the dual matrices of the new type. Theorem 4.. Let A a n and B b n be the dual matrices of the new type and μ any given sequence space. Then, A f : μ if and only if B f : μ and {( Rn )a n } r n n N c for every fixed N. Proof. Suppose that A a n and B b n are dual matrices of the new type, that is to say 4.2 holds and μ is an any given sequence space. Since the spaces f and f are linearly isomorphic, now let A f : μ and y y f. Then, BR exists and a n N d 2 cs,

12 2 Abstract and Applied Analysis which yields that b n N l for each n N. Hence, By exists for each y f, andthus letting m in the equality m m m b n y 0 0 j r j b nj x 4.4 for all m, n N, we have by 4.2 that By Ax, which gives the result B f : μ. Conversely, let {a n } N f β for each n N and B f : μ hold, and tae any x x f. Then, Ax exists. Therefore, we obtain from the equality m m a n x R Δ a n y a nm r 0 0 r n m b n y ; n N, 0 y m 4.5 as that Ax By, and this shows that A f : μ. This completes the proof. Theorem 4.2. Suppose that the entries of the infinite matrices D d n and E e n are connected with the relation e n r j d j, n, N 4.6 and μ is any given sequence space. Then, D μ : f if and if only E μ : f. Proof. Let x x μ, and consider the following equality with 4.6 : r j R j m m d j x e n x ; m, n, N, which yields as m that Dx f whenever x μ if and if only Ex f whenever x μ. This step completes the proof. Now, right here, we give the following propositions that are obtained from Lemmas 3.2 and 3. and Theorems 4. and 4.2. Proposition 4.3. Let A a n be an infinite matrix of real or complex numbers. Then, ( ) A a n f : l Δ r j a nj 2 {a n } f β n N. j a 0, 4.8

13 Abstract and Applied Analysis 3 Proposition 4.4. Let A a n be an infinite matrix of real or complex numbers. Then, 3 r j a nj a, r j a nj ( ) 4 a for each N, R A a n f : c j j 5 Δ r j a nj a R 0, j j 6 {a n } N f β n N. j 4.9 Proposition 4.5. Let A a n be an infinite matrix of real or complex numbers. Then, A a n ( ) l : f n N m r j a nj <, j r j a nj f α exists for each fixed N, R j j m r j a n i,j α m R 0 uniformly in n. j j 4.0 Proposition 4.6. Let A a n be an infinite matrix of real or complex numbers. Then, A a n ( ) c : f 0 2 n N r j a nj <, f j j f r j a nj α exists for each fixed N, j r j a nj α Core Theorems In this section, we give some core theorems related to the space f. We need the following lemma due to Das 25 for the proof of next theorem. Lemma 5.. Let c c nj p < and p N c nj p 0. Then, there is a y y j l such that y and p N j c nj ( p ) yj ( ) cnj p. p N j 5.

14 4 Abstract and Applied Analysis Theorem 5.2. B R core Ax K core x for all x l if and only if A c : f reg and p N n R j r i a i p,. 5.2 Proof. Necessity: Suppose first that B R core Ax K core x for all x l.ifx f, then we have τ Ax τ Ax. By this hypothesis, we get L x τ Ax τ Ax L x. 5.3 If x c, then L x L x x. So, we have f Ax τ Ax τ Ax x, which implies that A c, f reg. Now, let us consider the sequence C c nj p of infinite matrices defined by c nj ( p ) n R j r i a i p, n, i, p N. 5.4 Then, it is easy to see that the conditions of Lemma 5. are satisfied for the matrix sequence C. Thus, by using the hypothesis, we can write inf p N j cnj ( p ) p N p N ( ) cnj p j ( ) c nj p yj τ ( Ay ) L ( y ) y. j 5.5 This gives the necessity of 5.2. Sufficiency: Conversely, let A c : f reg and 5.2 hold for all x l. For any real number λ we write λ max{ λ, 0} and λ max{ λ, 0}; then λ λ λ and λ λ λ. Therefore, for any given ε>0, there is a j 0 N such that x j <L x ε for all j>j 0.Now,we can write ( ) c nj p xj ( ) c nj p xj ( ( )) xj cnj p ( ( )) xj cnj p j<j 0 j j 0 j j 0 j x ( ) ( ) cnj p L x ε cnj p j<j 0 x j 5.6 j [ ( ) ( )] cnj p cnj p. Thus, by applying p N and using the hypothesis, we have τ Ax L x ε. This completes the proof since ε is arbitrary and x l. In particular r i for all i since R is reduced to Cesàro matrix, see 8.

15 Abstract and Applied Analysis 5 Theorem 5.3. B C core Ax K core x for all x l if and only if A c : f reg and p N n i 0 a,i. 5.7 Theorem 5.4. A S l : f reg if and only if A c : f reg and n E R j r i a i p, 0 uniformly in p 5.8 for every E N with natural density zero. Proof. Necessity: Let A S l, f reg. Then, A c, f reg immediately follows from the fact that c S l. Now, define a sequence t t for x l as x, E, t 0, / E, 5.9 where E is any subset of N with δ E 0. Then, st t 0andt S 0, and so we have At f 0. On the other hand, since At E a nt, the matrix B b n defined by a n, E, b n 0, / E, 5.0 for all n, must belong to the class l, f 0. Hence, the necessity of 5.8 follows from Proposition 4.5. Sufficiency: Conversely, pose that A c, f reg and 5.8 holds. Let x S l and st x l.writee { : x l ε} for any given ε>0sothatδ E 0. Since A c, f reg and f a n, we have ( f Ax f a n x l l ) a n ( ) f a n x l l 5. p N n R j r i a i p, x l l.

16 6 Abstract and Applied Analysis On the other hand, since n R j r i a i p, x l x n E R j r i a i p, ε A, 5.2 condition 5.8 implies that n R j r i a i p, x l 0 uniformly in p. 5.3 Hence, f Ax st x; thatis,a S m, f reg, which completes the proof. Similarly, r i for all i since R is reduced to Cesàro matrix, see 8. Theorem 5.5. A S l : f reg if and only if A c : f reg and n i E 0 a,i 0 uniformly in p 5.4 for every E N with natural density zero. Theorem 5.6. B R core Ax st core x for all x l if and only if A S l : f reg and 5.2 holds. Proof. Necessity: Let B R core Ax st core x for all x l. Then, τ Ax β x for all x l, where β x st x. Hence, since β x st x L x for all x l see 3, we have 5.2 from Theorem 5.2. Furthermore, one can also easily see that β x τ Ax τ Ax β x, thatis, st inf x τ Ax τ Ax st x. 5.5 If x S l, then st inf x st x st x. Thus, the last inequality implies that st x τ Ax τ Ax f Ax, thatis,a S l : f reg. Sufficiency: Conversely, assume that A S l : f reg and 5.2 hold. If x l, then β x is finite. Let E be a subset of N defined by E {l : x i >β x ε} for a given ε>0. Then it is obvious that δ E 0andx i β x ε if l/ E.

17 Abstract and Applied Analysis 7 For any real number λ we write λ max{ λ, 0} and λ max{ λ, 0} whence λ λ λ, λ λ λ and λ λ 2 λ. Now, we can write ( ) c nj p xj ( ) c nj p xj ( ) c nj p xj j<j 0 j j 0 j ( ) c nj p xj c ( ) nj p xj c ) nj( p xj j<j 0 j j 0 j j0 x ( ) cnj p c ( ) nj p xj c ) nj( p xj j<j 0 j j 0 j j0 j/ E j E x [ ( ) ( )] ( ) c nj p c nj p x c nj p j j 0 j<j 0 [ β x ε ] ( ) ( ) cnj p x cnj p j j 0 j/ E j j 0 j E 5.6 x j j 0 [ c nj ( p ) c nj ( p )]. By applying the operator p N and using the hypothesis, we obtained that τ Ax β x ε. Since ε is arbitrary, we conclude that τ Ax β x for all x l,thatis, B R core Ax st core x for all x l and the proof is complete. Now if r i for all i, then R is reduced to Cesàro matrix and we have B C core Ax st core x, x l if and only if A ( ) S l : f reg 5.7 and 5.7 holds, see 8. Acnowledgment The authors would lie to than the referees for much constructive criticism and attention for details. References J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, UK, B. Altay and F. Başar, The fine spectrum and the matrix domain of the difference operator Δ on the sequence space l p, 0 <p<, Communications in Mathematical Analysis, vol. 2, no. 2, pp., M. Başarir, On some new sequence spaces and related matrix transformations, Indian Journal of Pure and Applied Mathematics, vol. 26, no. 0, pp , C. Aydın andf.başar, Some generalizations of the sequencespace a p r, Iranian Journal of Science and Technology, vol. 20, no. 2, pp , M. Kirişçi and F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Computers & Mathematics with Applications, vol. 60, no. 5, pp , M. Şengönül and F. Başar, Some new Cesàro sequence spaces of non-absolute type which include the spaces c 0 and c, Soochow Journal of Mathematics, vol. 3, no., pp. 07 9, 2005.

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