On some I-convergent sequence spaces defined by a compact operator

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1 Annals of the University of Craiova, Mathematics and Computer Science Series Volume 43(2), 2016, Pages ISSN: On some I-convergent sequence spaces defined by a compact operator Vaeel A. Khan, Mohd Shafiq, Rami Kamel Ahmad Rababah, and Ayhan Esi Abstract. In this article we introduce and study I-convergent sequence spaces S I, S I 0 and S I with the help of a compact operator T on the real space R. We study some topological and algebraic properties, prove the decomposition theorem and study some inclusion relations on these spaces. Key words and phrases. Compact operator, Ideal, filter, I-convergent sequence, solid and monotone space, Banach space, Lipschitz function. 1. Introduction and preliminaries Let N, R and C be the sets of all natural, real and complex numbers respectively. We denote ω = x = (x ) : x R or C the space of all real or complex sequences. Definition 1.1. Let K be a non-trivial scalar valued field and X be a vector space over K. Then a real valued mapping. on X is said to be a norm on or over X if it satisfies the following properties: (1) x 0, x = 0 x = 0, (2) αx = α x, (3) x + y x + y, for all α K, x, y X. The pair (X,. ) is called a normed linear space over K. Remar 1.1. If K = R (field of reals) or K = C ( field of complex), then X is called a real/complex normed linear space respectively. Definition 1.2. A normed linear space X is said to be a Banach space if it is complete. That is, if every Cauchy sequence in X is convergent in X. Definition 1.3. A linear operator T is an operator such that (1) the domain D(T ) of T is a vector space and the range R(T ) lies in a vector space over the same field, (2) T (αx + βy) = αt (x) + βt (y), for all, x, y D(T ). Received December 15,

2 142 V.A. KHAN, M. SHAFIQ, R.K.A. RABABAH, AND A. ESI Definition 1.4. Let X and Y be two normed linear spaces and T : D(T ) Y be a linear operator, where D(T ) X. Then, the operator T is said to be bounded if there exists a real > 0 such that T x x, for all, x D(T ). The set of all bounded linear operators B(X, Y ) is a normed linear space normed by T = sup T x (see [9, 10]) x X, x =1 and B(X, Y ) is a Banach space if Y is Banach space. Definition 1.5. Let X and Y be two normed linear spaces. An operator T : X Y is said to be a compact linear operator (or completely continuous linear operator) if (1) T is linear, (2) T maps every bounded sequence (x ) in X onto a sequence T (x ) in Y which has a convergent subsequence. The set of all compact linear operators C(X, Y ) is closed subspace of B(X, Y ) and C(X, Y ) is a Banach space if Y is Banach space. Throughout the paper, we denote l, c and c 0 as the Banach spaces of bounded, convergent and null sequences of reals respectively with norm x = sup x. Following Basar and Altay [1] and Sengönül [15], we introduce the sequence spaces S and S 0 with the help of compact operator T on R as follows. S = x = (x ) l : T x c and S 0 = x = (x ) l : T x c 0. As a generalisation of usual convergence, the concept of statistical convergent was first introduced by Fast [3] and also independently by Buc [2] and Schoenberg [14] for real and complex sequences. Later on, it was further investigated from a sequence space point of view and lined with the Summability Theory by Fridy 4], Šalát[11], Tripathy [16] and many others. Definition 1.6. A sequence x=(x ) ω is said to be statistically convergent to a limit L if for every ɛ > 0, we have lim 1 n N : x n L ɛ, n = 0. where vertical lines denote the cardinality of the enclosed set. That is, if δ(a(ɛ)) = 0, where A(ɛ) = N : x L ɛ. The notion of ideal convergence (I-convergence) was introduced and studied by Kostyro, Mačaj, Salǎt and Wilczyńsi [7,8]. Later on, it was studied by Šalát, Tripathy and Ziman [12,13], Tripathy and Hazaria [17,18], Khan and Ebadullah [5,6] and many others.

3 ON SOME I-CONVERGENT SEQUENCE SPACES Here we give some preliminaries about the notion of I-convergence. Definition 1.7. Let N be a non empty set. Then a family of sets I 2 N (power set of N) is said to be an ideal if (1) I is additive i.e A, B I A B I, (2) I is hereditary i.e A Iand B A B I. Definition 1.8. A non-empty family of sets J 2 N is said to be filter on N if and only if (1) Φ / J, (2) A, B J A B J, (3) A J and A B B J. Definition 1.9. An Ideal I 2 N is called non-trivial, if I =2 N. Definition A non-trivial ideal I 2 N is called admissible, if x : x N I. Definition A non-trivial ideal I is maximal, if there cannot exist any non-trivial ideal J =I containing I as a subset. Remar 1.2. For each ideal I, there is a filter (I) corresponding to I. That is (I) = K N : K c I, where K c = N \ K. Definition A sequence x = (x ) ω is said to be I-convergent to a number L, if for every ɛ > 0, the set N : x L ɛ I. In this case, we write I lim x = L. Definition A sequence x = (x ) ω is said to be I-null, if L = 0. In this case, we write I lim x = 0. Definition A sequence x = (x ) ω is said to be I-cauchy, if for every ɛ > 0 there exists a number m = m(ɛ) such that N : x x m ɛ I. Definition A sequence x = (x ) ω is said to be I-bounded, if there exists some M > 0 such that N : x M I. Definition A sequence space E is said to be solid(normal), if (α x ) E whenever (x ) E and for any sequence(α ) of scalars with α 1, for all N. Definition A sequence space E is said to be symmetric, if (x π() ) E whenever x E, where π is a permutation on N. Definition A sequence space E is said to be sequence algebra, if (x ) (y ) = (x.y ) E whenever (x ), (y ) E. Definition A sequence space E is said to be convergence free, if (y ) E whenever (x ) E and x = 0 implies y = 0, for all. Definition Let K = 1 < 2 < 3 < 4 < 5... N and E be a Sequence space. A K-step space of E is a sequence space λ E K = (x n ) ω : (x ) E. Definition A canonical pre-image of a sequence (x n ) λ E K (y ) ω defined by x, if K, y = 0, otherwise. is a sequence

4 144 V.A. KHAN, M. SHAFIQ, R.K.A. RABABAH, AND A. ESI A canonical preimage of a step space λ E K is a set of preimages of all elements in λ E K.i.e. y is in the canonical preimage of λe K, iff y is the canonical preimage of some x λ E K. Definition A sequence space E is said to be monotone, if it contains the canonical preimages of its step space. Definition If I = I f, the class of all finite subsets of N. Then, I is an admissible ideal in N and I f convergence coincides with the usual convergence. Definition If I = I δ = A N : δ(a) = 0. Then, I is an admissible ideal in N and we call the I δ -convergence as the logarithmic statistical convergence. Definition If I = I d = A N : d(a) = 0. Then, I is an admissible ideal in N and we call the I d -convergence as the asymptotic statistical convergence. Remar 1.3. If I δ lim x n = l, then I d lim x n = l. Definition A map defined on a domain D X i.e : D X IR is said to satisfy Lipschitz condition if (x) (y) K x y where K is nown as the Lipschitz constant. Definition A convergence field of I-covergence is a set F (I) = x = (x ) l : there exists I lim x IR. The convergence field F (I) is a closed linear subspace of l with respect to the supremum norm, F (I) = l c I (see[12]). The function : F (I) R defined by (x) = I lim x, for all x F (I) is a lipschitz function (see[12]). We used the following lemmas to establish some results of this article. Lemma(I). Every solid space is monotone. Lemma(II). Let K (I) and M N. If M / I, then M K / I. Lemma(III). If I 2 N and M N. If M / I, then M N / I. Throughout the article, T is considered as a compact operator on the real space R. 2. Main Results In this article we introduce and study the following classes of sequence. S I = x = (x ) l : N : T (x ) L ɛ I, for some L R ; (2.1) S I = S I 0 = x = (x ) l : N : T (x ) ɛ I ; (2.2) x = (x ) l : M > 0 s.t. N : T (x ) M I. (2.3) Theorem 2.1. The classes of sequences S I, S I 0 and S I are linear spaces.

5 ON SOME I-CONVERGENT SEQUENCE SPACES Proof. We shall prove the result for the space S I. Rests will follow similarly. For, let x = (x ), y = (y ) be two elements of S I and α, β be scalars. Now, since (x ), (y ) S I, then, for given ɛ > 0, there exists L 1, L 2 R such that the sets and Therefore, N : T (x ) L 1 < ɛ (I) (2.4) 2 α N : T (y ) L 2 < ɛ (I). (2.5) 2 β T ( α(x ) + β(y ) ) ( αl 1 + βl 2 ) = αt (x ) + βt (y ) ( αl 1 + βl 2 ) = αt (x ) αl 1 + βt (y ) βl 2 α T (x ) L 1 + β T (y ) L 2 < α ɛ 2 α + β ɛ 2 β = ɛ 2 + ɛ 2 = ɛ. Thus the set A 3 = N : T ( α(x ) + β(y ) ) ( ) αl 1 + βl 2 < ɛ (I). Therefore the set A c 3 = N : T ( α(x ) + β(y ) ) ( ) αl 1 + βl 2 ɛ I implies that α(x ) + β(y ) S I, for all scalars α, β and (x ), (y ) S I. Hence S I is linear. Theorem 2.2. The spaces S I and S I 0 are normed spaces normed by x = sup T (x ). Proof. The proof of the result is easy in view of existing techniques and hence omitted. Theorem 2.3. A sequence x = (x ) l I-converges if and only if for every ɛ > 0, there exists N ɛ N such that N : T (x ) T (x Nɛ ) < ɛ (I). (2.6) Proof. Let x = (x ) l. Suppose that L = I lim x. Then, the set B ɛ = N : T (x ) L < ɛ (I) for all ɛ > 0. 2 Fix an N ɛ B ɛ. Then, we have T (x ) T (x Nɛ ) T (x ) L + T (x Nɛ ) L < ɛ 2 + ɛ 2 = ɛ

6 146 V.A. KHAN, M. SHAFIQ, R.K.A. RABABAH, AND A. ESI which holds for all B ɛ. Hence N : T (x ) T (x Nɛ ) < ɛ (I). Conversely, suppose that N : T (x ) T (x Nɛ ) < ɛ (I). That is N : T (x ) T (x Nɛ ) < ɛ (I), for all ɛ > 0. Then, the set C ɛ = N : T (x ) [T (x Nɛ ) ɛ, T (x Nɛ ) + ɛ] (I) for all ɛ > 0. [ ] Let J ɛ = T (x Nɛ ) ɛ, T (x Nɛ ) + ɛ. If we fix an ɛ > 0 then we have C ɛ (I) as well as C ɛ (I). Hence C 2 ɛ C ɛ (I). This implies that 2 J = J ɛ J ɛ φ. 2 That is N : T (x ) J (I). That is diamj diamj ɛ where the diam of J denotes the length of interval J. In this way, by induction, we get the sequence of closed intervals J ɛ = I 0 I 1... I... with the property that diami 1 2 diami 1 for (=2,3,4,...) and N : T (x ) I (I) for (=1,2,3,4,...). Then there exists a ξ I where N such that ξ = I lim T (x ). Hence the result. Theorem 2.4. Let I be an admissible ideal. Then, the following are equivalent. (a) (x ) S I ; (b) there exists (y ) S such that x = y, for a.a..r.i; (c) there exists (y ) S and (z ) S I such that x = y + z for all N and N : T (y ) L ɛ I; (d) there exists a subset K = 1 < 2 < 3... of N such that K (I) and lim n ) L = 0. n Proof. (a) implies (b). Let (x ) S I. Then, for any ɛ > 0, there exists L R such that the set N : T (x ) L ɛ I. Let (m t ) be an increasing sequence with m t N such that Define a sequence (y ) as For m t < m t+1, t N. m t : T (x ) L t 1 I. y = y = x, for all m 1. x, if T (x ) L < t 1, L, otherwise.

7 ON SOME I-CONVERGENT SEQUENCE SPACES Then, (y ) S and from the following inclusion m t : x y N : T (x ) L ɛ I. We get x = y, for a.a..r.i. (b) implies (c). For (x ) S I. Then, there exists (y ) S such that x = y, for a.a..r.i. Let K = N : x y, then K I. Define a sequence (z ) as x y z =, if K, 0, otherwise. Then (z ) S I and (y ) S. (c) implies (d). Let P 1 = N : T (x ) ɛ I and K = P c 1 = 1 < 2 < 3 <... (I). Then, we have lim n T (x n ) L = 0. (d) implies (a). Let K = 1 < 2 < 3 <... (I) and lim T (x n ) L = 0. n Then, for any ɛ > 0, and by Lemma (II), we have Thus, (x ) S I. N : T (x ) L ɛ K c K : T (x ) L ɛ. Theorem 2.5. The function : S I R defined by (x) = I lim T (x), for all x S I is a Lipschitz function and hence uniformly continuous. Proof. Clearly the function is well defined. Let x = (x ), y = (y ) S I, x y. Then, the sets A x = N : T (x) (x) x y I. A y = N : T (y) (y) x y I. where x y = sup T (x y ). Thus, the sets B x = N : T (x) (x) < x y (I). B y = N : T (y) (y) < x y (I). Hence, B = B x B y (I), so that B. Now taing B, we have (x) (y) (x) T (x) + T (x) T (y) + T (y) (y) 3 x y. Thus, is Lipschitz function and hence uniformly continuous. Theorem 2.6. If T is an identity operator and : S I R is a function defined by (x) = I lim T (x), for all x S I and if x = (x ), y = (y ) S I, then, (x.y) S I and (x.y) = (x) (y). Proof. For ɛ > 0, the sets B x = N : T (x) (x) < ɛ (I), (2.7) B y = N : T (y) (y) < ɛ (I). (2.8) where x y = ɛ. Now, since T is an identity operator, we have T (xy) (x) (y) = T (x y ) (x) (y) = T (x y ) T (x ) (y)+t (x ) (y) (x) (y)

8 148 V.A. KHAN, M. SHAFIQ, R.K.A. RABABAH, AND A. ESI = x y x (y) + x (y) (x) (y) x y (y) + (y) x (x). (2.9) As S I l, there exists an M R such that x < M and (y) < M. Therefore, from (2.7), (2.8) and (2.9), we have T (xy) (x) (y) = T (x y ) (x) (y) Mɛ + Mɛ = 2Mɛ for all B x B y (I). Hence (x.y) S I and (x.y) = (x) (y). Theorem 2.7. The space S I is solid and monotone. Proof. For, let (x ) S I. Then, the set N : T (x ) ɛ I. (2.10) Let (α ) be a sequence of scalars with α 1, for all, N. Therefore, T (α x ) = α T (x ) α T x T x, for all N. Thus, from the above inequality and (2.10), we have N : T (α x ) ɛ N : T (x ) ɛ I implies that N : T (α x ) ɛ I. Therefore, α x S I. Hence the space S I is solid. That the space is monotone follows from lemma (I). Theorem 2.8. The inclusions S I 0 S I S I hold. Proof. Let (x ) S I. Then, there exists some L such that I lim T (x ) L = 0. That is, the set N : T (x ) L ɛ I. We have T (x ) = T (x ) L + L T (x ) L + L. Taing supremum over on both sides, we get (x ) S. I The inclusion S0 I S I is obvious. Hence S I S I S. I Theorem 2.9. The set S I is closed subspace of l. Proof. Let (x (n) ) be a Cauchy sequence in SI such that x (n) x. We show that x S I. Since (x (n) ) SI, then there exists a n such that N : T (x (n) ) a n ɛ I. We need to show that (1) (a n ) converges to a. (2) If U = N : T (x ) a < ɛ, then U c I. (1) Since (x (n) ) is Cauchy sequence in SI for a given ɛ > 0, there exists 0 N such that sup T (x (n) ) T (x(q) ) < ɛ 3, for all n, q 0. For a given ɛ > 0, we have B nq = N : T x (n) T x (q) < ɛ 3

9 ON SOME I-CONVERGENT SEQUENCE SPACES B q = N : T x (q) a q < ɛ 3 B n = N : T x (n) a n < ɛ 3 Then, Bnq, c Bq, c Bn c I. Let B c = Bnq c Bq c Bn, c where B = N : a q a δ n < ɛ. Then, B c I. We choose 0 B c. Then for each n, q 0, we have [ N : a q a n < ɛ N : a q T (x (q) ) < ɛ 3 N : T (x (q) ) T (x(n) ) < ɛ N : T (x(n) 3 ) a n < ɛ ] 3. Then (a n ) is a Cauchy sequence of scalars in R, so there exists a scalar a in R such that a n a as n. (2) Let 0 < δ < 1 be given. Then we show that if U = N : T (x ) a δ, then U c I. Since x (n) x, then there exists q 0 N such that P = N : T (x (q0) ) T (x ) < δ (2.11) 3 implies P c I. The number q 0 can be chosen that together with (2.11), we have Q = N : a q0 a < δ 3 such that Q c I. Since N : T (x (q0) ) a q0 δ I. Then we have a subset S of N such that S c I, where S = N : T (x (q0) ) a q0 < δ 3. Let U c = P c Q c S c, where U = N : T (x ) a < δ. Therefore, for each U c, we have N : T (x ) a < δ Then the result follows. [ N : T (x ) T (x (q0) ) < δ 3 N : T (x (q0) ) a q0 < δ 3 N : a q 0 a p < δ 3 ]. Since the inclusions S I l and is strict so in view of Theorem (2.9.) we have the following result. Theorem The space S I is nowhere dense subsets of l. References [1] F. Basar, B. Altay, On the spaces of sequences of p-bounded variation and related matrix mapping, Urainian Math. J. 55 (2003). [2] R.C. Buc, Generalized Asymptotic Density, Amer.J.Math. 75 (1953), [3] Fast, Sur la convergence statistique, Colloq.Math H. 2 (1951), [4] J.A. Fridy, On statistical convergence, Analysis 5 (1985), [5] V.A. Khan, K. Ebadullah, On Zweier I-convergent sequence spaces, (in press).

10 150 V.A. KHAN, M. SHAFIQ, R.K.A. RABABAH, AND A. ESI [6] V.A. Khan, K. Ebadullah, On some I-Convergent sequence spaces defined by a modullus function, Theory and Applications of Mathematics and Computer Science 1 (2011), no. 2, [7] P. Kostyro, M. Ma caj, T. Šalát, M. Silezia, I-convergence and extremal I-limit points, Mathematica Slovaca 55 (2005), no. 4, [8] P. Kostyro, T. Šalát, W. Wilczyńsi, I-convergence, Real Analysis Analysis Exchange 26 (2000), no. 2, [9] E. Kreyszig, Introductory functional analysis with applications, John Wiley and Sons Inc, New Yor-Chichester-Brisbane-Toronto, [10] I.J. Maddox, Elements of functional analysis,cambridge University Press, [11] T. Šalát, On statistical convergent sequences of real numbers, Math. Slovaca 30 (1980). [12] T. Šalát, B.C. Tripathy, M. Ziman, On some properties of I-convergence, Tatra Mt. Math. Publ. 28 (2004), [13] T. Šalát, B.C. Tripathy, M. Ziman, On I-convergence field, Ital. J. Pure Appl. Math. 17 (2005), [14] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), [15] M. Sengönül, The Zweier sequence space, Demonstratio Mathematica XL (2007), no.1, [16] B.C. Tripathy, On statistical convergence, Proc. Estonian Acad. Sci. Phy. Math. Analysis (1998), [17] B.C. Tripathy, B. Hazaria, Some I-Convergent sequence spaces defined by Orlicz function, Acta Mathematicae Applicatae Sinica 27 (2011), no. 1, [18] B.C. Tripathy, B. Hazaria, Paranorm I-convergent sequence spaces, Math. Slovaca 59 (2009), no. 4, (Vaeel A. Khan) Department of Mathematics A.M.U, Aligarh , India address: vahanmaths@gmail.com (Mohd Shafiq) Department of Mathematics A.M.U, Aligarh , India address: shafiqmaths7@gmail.com (Rami Kamel Ahmad Rababah) Department of Mathematics A.M.U, Aligarh , India address: ramirababahr@hotmail.com (Ayhan Esi) Adiyaman University, Science and Art Faculty, Department of Mathematics, Turey address: aesi23@hotmail.com