/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 3, 2015, 357-367 ISSN 1307-5543 – www.ejpam.com On Statistical and Ideal Convergence of Sequences of Bounded Linear Operators Enno Kolk Institute of Mathematics, University of Tartu, 50090 Tartu, Estonia Abstract. Let (An) be a sequence of bounded linear operators from a separable Banach space X into a Banach space Y . Suppose that Φ is a countable fundamental set of X and the ideal I of subsets of N has property (AP). The sequence (An) is said to be b∗I -convergent if it is pointwise I -convergent and there exists an index set K such that N \ K ∈ I and (Ak x)k∈K is bounded for any x ∈ X . We prove that the sequence (An) is b∗I -convergent if and only if (‖An‖) is I -bounded and (Anφ) is I -convergent for any φ ∈ Φ. Applications of this Banach–Steinhaus type theorem are related to some sequence-to-sequence matrix transformations and to the weak I -convergence in Banach spaces. 2010 Mathematics Subject Classifications: 40A35; 40C05, 40J05, 46B15, 46B45 Key Words and Phrases: Banach space, bounded linear operator, ideal, I -convergence, I -boundedness, matrix method of summability, sequence space, statistical convergence, weak I -convergence, weak statistical convergence 1. Introduction and Preliminaries Let N = {1,2, . . . } and let X , Y be two normed spaces over the field K of real numbers R or complex numbers C. A subset Φ of X is called fundamental if the linear span of Φ is dense in X . By B(X , Y ) we denote the space of all bounded linear operators from X into Y . As usual, the dual of X is defined by X ′ = B(X ,K). Byω(X )we denote the set of all X -valued sequences. We write supn, limn and ∑ n instead of supn∈N, limn→∞ and ∑∞ n=1, respectively. By an index set we mean any infinite set {ki} ⊂ N with ki < ki+1 for each i ∈ N. Let An ∈ B(X , Y ) (n ∈ N). The following theorems of functional analysis are well known (see, for example, [11] or [17]). Theorem 1 (Principle of uniform boundedness). Let X be a Banach space. If supn ‖An x‖<∞ for every x ∈ X , then sup n ‖An‖<∞. (1) Email address: enno.kolk@ut.ee http://www.ejpam.com 357 c© 2015 EJPAM All rights reserved. E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 358 Theorem 2 (Banach–Steinhaus). Let X , Y be two Banach spaces and let Φ be a fundamental set of X . The limit limn An x exists for any x ∈ X if and only if (1) holds and limn Anφ exists for every φ ∈ Φ. Moreover, the limit operator A0, A0 x = limn An x is bounded and linear, i.e., A0 ∈ B(X , Y ), and ‖A0‖ ≤ supn ‖An‖. If A ∈ B(X , Y ), then limn An x = Ax for any x ∈ X if and only if (1) holds and limn Anφ = Aφ (φ ∈ Φ). The first idea of statistical convergence appeared, under the name of almost convergence, in the first edition (Warsaw, 1935) of the monograph [25] of Zygmund. Since 1951 when Fast [7] (see also [23] and [22]) introduced statistical convergence of number sequences in terms of asymptotic density of subsets of N, several applications and generalizations of this notion have been investigated (for references see [4] and [6]). For instance, Maddox [20] and Kolk [13] considered the statistical convergence of sequences taking values in a locally convex space or a normed space, respectively. An another extension of statistical convergence is related to generalized densities. Let T = (tnk) be a non-negative regular matrix of scalars (i.e., tnk ≥ 0 (n, k ∈ N) and limn ∑ k tnkuk = limk uk for any convergent scalar sequence (uk)). A set K ⊂ N is said to have T-density δT (K) if the limit δT (K) = lim n ∑ k∈K tnk exists (cf. [9]). A sequence x = (xk) ∈ ω(X ) is called T-statistically convergent to a point l ∈ X , briefly stT -lim xk = l, if δT ({k : ‖xk − l‖ ≥ ǫ}) = 0 for every ǫ > 0 (see [3, Definition 7] and [14, p. 44]). If T is the identity matrix I , then T -statistical convergence is just the ordinary convergence in X and if T is the Cesàro matrix C1, then T -statistical convergence is statistical convergence as defined by Fast [7]. A further extension of statistical convergence was given in [16] by means of ideals. Recall that a subfamily I of the family 2N of all subsets of N is called an ideal if for each K , L ∈ I we have K ⋃ L ∈ I and for each K ∈ I and each L ⊂ K we have L ∈ I . An ideal I is called non-trivial if I 6= ; and N /∈ I . A non-trivial ideal I is called admissible if I contains all finite subsets of N. Any non-trivial ideal I defines a filter F (I ) = {K ⊂ N : N \ K ∈ I }. For example, IT = {K ⊂ N : δT (K) = 0} is an admissible ideal and the IT -convergence coincides with the T -statistical convergence. An admissible ideal I ⊂ 2N is said to have property (AP) if for every countable family of mutually disjoint sets K1, K2, . . . from I there exist sets L1, L2, . . . from 2N such that the symmetric differences Ki∆Li (i ∈ N) are finite and L = ⋃ i Li ∈ I . E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 359 Remark 1 ([1], Proposition 1). The property (AP) is equivalent to the property (P): for every countable family of sets K1, K2, . . . from I there exist a set K ∈ I such that the differences Ki \K (i ∈ N) are finite. A sequence x = (xk) ∈ ω(X ) is said to be I -convergent to l ∈ X , briefly I -limk xk = l, if for each ǫ > 0 the set {k ∈ N : ‖xk − l‖ ≥ ǫ} belongs to I [16, Definition 3.1]. With the I -convergence are closely related the following two notions. A sequence x = (xk) ∈ ω(X ) is said to be I ∗-convergent to l ∈ X , briefly I ∗-lim xk = l, if there exists an index set K = (ki) such that K ∈ F (I ) and limi xki = l in X [16, Definition 3.2]). A sequence x = (xk) ∈ ω(X ) is said to be I -bounded, briefly xk = OI (1), if there exists an index set K = (ki) such that K ∈ F (I ) and the sequence (ki) is bounded in X (cf. [10]). In the special case I = IT we write OstT (1) instead of OI (1). We remark that theI ∗-convergence of number sequences was introduced already by Freed- man [8] as I -near convergence. It is easy to see that I ∗-convergence implies I -convergence and every I ∗-convergent sequence is I -bounded. The following characterization of I -convergence is important for us. Proposition 1 ([16, Theorem 3.2]). If the ideal I has property (AP), then I -lim xk = l in a Banach space X if and only if I ∗-lim xk = l. By cI (X ) we denote the set of all I -convergent X -valued sequences. Let ℓ∞(X ), c(X ) and c0(X ) be the sets of all bounded, convergent and convergent to zero X -valued sequences, respectively. For 1 ≤ p < ∞ let ℓp(X ) be the set of sequences (xk) ∈ ω(X ) such that ∑ k ‖xk‖ p <∞. Using Proposition 1 and Theorem 2, we proved in [15] the following Banach–Steinhaus type theorem for I -convergence. Theorem 3 ([15, Theorem 3]). Let X and Y be two Banach spaces, where X has a countable fundamental set Φ. If the ideal I has property (AP), then the sequence (An) is bI -convergent (i.e., (An x) ∈ cI (Y )∩ ℓ∞(Y ) for any x ∈ X ) if and only if (1) holds and (Anφ) is I -convergent for every φ ∈ Φ. Thereby, the limit operator A, Ax = I -lim An x, is bounded and linear, and ‖A‖ ≤ supn ‖An‖. In this paper we introduce the notion of b∗I -convergence of sequences of bounded lin- ear operators (An) and give an analogue of Theorem 3 by finding necessary and sufficient conditions for b∗I -convergence of such sequences (An). As applications of this result we char- acterize infinite summability matrices A = (Ank) of type A : λ(X ) b∗I −→ c(Y ) with Ank ∈ B(X , Y ) (n, k ∈ N) and λ ∈ {c, c0, ℓ1}, also consider the weak b∗I -convergence in Banach spaces. 2. Main Theorems In the following let X , Y be two Banach spaces, An ∈ B(X , Y ) (n ∈ N) and let I ⊂ 2N be a non-trivial admissible ideal. E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 360 Recall that a sequence (xn) ∈ω(X ) is said to be weakly I -convergent (weakly T-statistically convergent) to a point l ∈ X if I -lim x ′(xn) = x ′(l) (stT -lim x ′(xn) = x ′(l)) for any x ′ ∈ X ′ [2, 21]. We know that every weakly convergent sequence in a Banach space X is bounded. But a weakly I -convergent sequence is not necessary I -bounded (cf. [5, Theorem 1]). Example 2 from [5] shows that the Banach sequence space ℓ2 contains a weakly statistically null sequence (zk) with no bounded subsequences. Thus some results of Bhardwaj and Bala [2, Theorem 3.1 and Lemma 3.2] are incorrect. At it, defining Fn x ′ = x ′(zn) (x ′ ∈ ℓ′2, n ∈ N), we get the sequence (Fn) of bounded linear functionals Fn : ℓ′2 → R. Since ‖Fn‖ = ‖zn‖ by the classical Hahn–Banach theorem, the sequence of functionals (Fn) converges statistically to zero for any x ′ ∈ ℓ′2, but the sequence of norms (‖Fn‖) contains no bounded subsequences. This example justifies the following definition. Definition 1. A sequence (An) of operators An ∈ B(X , Y ) (n ∈ N) is said to be b∗I -convergent (to A∈ B(X , Y )) if I - limn An x exists (I - lim An x = Ax) for any x ∈ X and there is a set K ∈ F (I ) such that (Ak x)k∈K is bounded for every x ∈ X . In the special case I = IT we get the notion of b∗T-statistical convergence. The b∗I -limit and the b∗T-statistical limit of (An) are denoted, respectively, by b∗I - limn An and b∗stT - limn An. In view of Theorem 1 we can say that a sequence (An) is b∗I -convergent if and only if I -lim An x exists for any x ∈ X and sup k∈K ‖Ak‖)<∞ for some K ∈ F (I ). (2) Theorem 3 shows that bI -convergence implies b∗I -convergence by the suppositions that X is separable and I satisfies the condition (AP). To prove our main theorem we need the following lemma. Lemma 1. Suppose that the ideal I has property (AP) and let zk j ∈ X (k, j ∈ N). If I -limk zk j = z j for any j ∈ N, then there exists an index set N = (ni) such that N ∈ F (I ) and limi zni , j = z j for any j ∈ N. Proof. Assume that I -limk zk j = z j ( j ∈ N). Since I has property (AP), by Proposition 1 there exist index sets K j = {ki( j)} ( j ∈ N) such that lim i zki( j), j = z j ( j ∈ N) (3) and K ′ j = N \ K j ∈ I for any j ∈ N. Because of Remark 1 we can find the set N ′ ∈ I such that the differences K ′ j \ N ′ ( j ∈ N) are finite. Now, for N = N \ N ′ we have that N ∈ F (I ) and the differences N \K j are finite. Consequently, denoting N = (ni), from (3) it follows that limi zni , j = z j for any j ∈ N. Theorem 4. Let X and Y be two Banach spaces, where X has a countable fundamental set Φ. If the ideal I has property (AP). A sequence (An) of operators An ∈ B(X , Y ) is b∗I -convergent if and only if (‖An‖) is I -bounded, i.e., (2) holds, and (Anφ) is I -convergent for every φ ∈ Φ. Thereby, the limit operator A0, A0 x = I -lim An x, is bounded and linear, and ‖A0‖ ≤ supk∈K ‖Ak‖. If A ∈ B(X , Y ), then b∗I - limn An = A if and only if (‖An‖) is I -bounded and I - limn Anφ = Aφ (φ ∈ Φ). E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 361 Proof. If (An) is b∗I -convergent (b∗I - limn An = A), then (2) is satisfied and I -lim Anφ exists (I -lim Anφ = Aφ) for every φ ∈ Φ. Conversely, assume that (2) holds and I -lim Anφ j exists (or I -lim Anφ j = Aφ j) for every j ∈ N, where Φ = {φ j}. Applying Lemma 1 to zn j = Anφ j (and z j = Aφ j), we fix an index set N = (ni) ∈ F (I ) such that limi Ani φ j exists (limi Ani φ j = Aφ j) for any j ∈ N. Since the set M = N ∩ K also belongs to F (I ), denoting M = (mi), we have that limi Ami φ j exists (limi Ami φ j = Aφ j) for any j ∈ N and supi ‖Ami ‖<∞. So, by Theorem 2, the limit A0 x = limi Ami x exists (limi Ami x = Ax) for any x ∈ X , A0 ∈ B(X , Y ) and ‖A0‖ ≤ supi ‖Ami ‖. The proof is completed if we remark that limi Ami x = I - lim An x by Proposition 1. It is known that the ideal IT = {K ⊂ N : δT (K) = 0} defined by a non-negative regular matrix T has the property (AP) (see [9, Proposition 3.2]). Since IT -convergence coincides with T -statistical convergence, from Theorem 4 we immediately get the following Banach– Steinhaus type theorem for b∗T -statistical convergence. Theorem 5. Suppose that T is a non-negative regular matrix and X has a countable fundamental set Φ. A sequence (An) of operators An ∈ B(X , Y ) is b∗T-statistically convergent if and only if (2) holds and stT -lim Anφ exists for any φ ∈ Φ. In this case the limit operator A0, A0 x = stT -lim An x (x ∈ X ), belongs to B(X , Y ) and ‖A0‖ ≤ supk∈K ‖Ak‖. If A ∈ B(X , Y ), then b∗stT - limn An = A if and only if (‖An‖) is I -bounded and stT - limn Anφ = Aφ (φ ∈ Φ). 3. Some Applications Let λ(X ) be a subspace of ω(X ), µ(Y ) a subspaces of ω(Y ) and A = (Ank) an infinite matrix of operators Ank ∈ B(X , Y ) (n, k ∈ N). We say that A maps λ(X ) into µ(Y ), and write A : λ(X )→µ(Y ), if for all x = (xk) ∈ λ(X ) the series Anx = ∑ k Ank xk (n ∈ N) converge and the sequence Ax= (Anx) belongs to µ(Y ). It is well known that c(X ), c0(X ) and ℓ∞(X ) are Banach spaces with the norm ‖x‖∞ = supk ‖xk‖, and ℓp(X ) is Banach space with the norm ‖x‖p = �∑ k ‖xk‖ p �1/p if 1≤ p <∞. For x ∈ X and n ∈ N let e(x) = (x , x , . . . ) be constant sequence and ek(x) = (ek j (x)) the sequence with ek j (x) = x if j = k and ek j (x) = 0 otherwise. It is not difficult to see that if Φ is a (countable) fundamental set in X , then E0(Φ) = {e k(φ) : k ∈ N, φ ∈ Φ} is a (countable) fundamental set in Banach spaces c0(X ) and ℓp(X ), and E0(Φ) ⋃ E1(Φ) with E1(Φ) = {e(φ) : φ ∈ Φ} is a (countable) fundamental set in Banach space c(X ). Using Theorem 2, Zeller [24] (see also [19]) and Kangro [12] characterized the matrices A : c(X )→ c(Y ), A : c0(X )→ c(Y ) and A : ℓ1(X )→ c(Y ) as follows. Theorem 6. Let A= (Ank) be an infinite matrix with Ank ∈ B(X , Y ). Then: (i) A : c(X )→ c(Y ) if and only if Gn = sup r sup ‖xk‖≤1 r ∑ k=1 Ank xk <∞ (n ∈ N), (4) E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 362 sup n Gn <∞, (5) ∃ lim n Ank x (k ∈ N, x ∈ X ), (6) ∃ lim m m ∑ k=1 Ank x (n ∈ N, x ∈ X ), (7) ∃ lim n ∑ k Ank x (x ∈ X ); (8) (ii) A : c0(X )→ c(Y ) if and only if (4)–(6) hold; (iii) A : ℓ1(X )→ c(Y ) if and only if (6) is satisfied and Hn = sup k Ank <∞ (n ∈ N), (9) sup n Hn <∞, Remark 2. It is not difficult to see, using Theorem 2, that in Theorem 6 it suffices to require the fulfillment of conditions (6)–(8) for all elements φ from a fundamental set Φ of X . The notion of b∗I -convergence of sequences of bounded linear operators leads us to the definition of new type summability maps. Definition 2. Let λ(X ) and µ(Y ) be two linear subspaces of ω(X ) and ω(Y ), respectively, and let I ⊂ 2N be a non-trivial admissible ideal. We say that a matrix A maps λ(X ) in the sense of b∗I -convergence into µ(Y ), and write A : λ(X ) b∗I −→µ(Y ), if I - limAnx exists for any x ∈ λ(X ) and there is an index set N = (ni) from F (I ) such that the submatrix A(N) = (ani ,k) maps λ(X ) into ℓ∞(Y ). In the case of I = IT we get the matrices of type A : λ(X ) b∗stT −→µ(Y ). Based on Theorems 4 and 5, we describe the matrices A : λ(X ) b∗I −→ c(Y ) and A : λ(X ) b∗stT −→ c(Y ), where λ ∈ {c, c0, ℓp}. Proposition 2. Let A = (Ank) be an infinite matrix with Ank ∈ B(X , Y ). Suppose that X has a countable fundamental set Φ and the ideal I has property (AP). Then: (i) A : c(X ) b∗I −→ c(Y ) if and only if (4) and (7) hold, Gn =OI (1), (10) ∃I - lim n Ankφ (k ∈ N, φ ∈ Φ), (11) ∃I - lim n ∑ k Ankφ (φ ∈ Φ); (12) (ii) A : c0(X ) b∗I −→ c(Y ) if and only if (4), (10) and (11) hold; E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 363 (iii) A : ℓ1(X ) b∗I −→ c(Y ) if and only if (9) is satisfied and (Hn) is I -bounded. Proof. The equality A(r)n x = ∑r k=1 Ank xk defines a linear operator A(r)n on c(X ) and c0(X ) for any n, r ∈ N. Since ‖A(r)n ‖= r ∑ k=1 Ank xk , by Theorem 2 we get that the series Anx (n ∈ N) converge for all x ∈ c(X ) and An ∈ B(c(X ), Y ) if and only if (4), (7) are satisfied. Similarly, An ∈ B(c0(X ), Y ) if and only if (4) holds. Now, applying Theorem 4 to the operators An, we have that A : c(X ) b∗I −→ c(Y ) (or A : c0(X ) b∗I −→ c(Y )) if and only if (10) holds and (Any) is I -convergent for any y ∈ E1(Φ) (respectively, y ∈ E0(Φ)). But this reduces to (11) and (12) because Anek(φ) = Ankφ and Ane(φ) = ∑ k Ankφ. Since An ∈ B(ℓ1(X ), Y ) if and only if (9) holds, the statement (iii) also follows by Theo- rem 4. The matrix map A : ℓp(X ) b∗I −→ c(Y )we consider in the special cases Y = K and 1< p <∞. Then B(X , Y ) = X ′ and so, Ank ∈ X ′ (n, k ∈ N). In this case An ∈ (ℓp(X )) ′ if and only if (Ank)k∈N ∈ ℓq(X ′), i.e., ∑ k ‖Ank‖ q <∞, where 1/p + 1/q = 1. Therefore, denoting c = c(K) and using the same arguments as in the proof of Proposition 2, we get the following result. Proposition 3. Let A = (Ank) be an infinite matrix with Ank ∈ X ′. Suppose that X has a countable fundamental set Φ, the ideal I has property (AP) and 1 < p <∞, 1/p + 1/q = 1. Then A : ℓp(X ) b∗I −→ c if and only if (11) holds and ∑ k ‖Ank‖ q = OI (1). If X = Y = K, then the matrix map A reduces to the transformation A : λ→ µ defined by an infinite scalar matrix A= (ank). Using the fact that for Y = K we have (see [12, p. 114]) sup ‖xk‖≤1 r ∑ k=1 Ank xk = r ∑ k=1 ‖Ank‖, from Propositions 2 and 3 we obtain the following corollary. Corollary 1. Let A= (ank) be an infinite matrix of scalars, 1 < p <∞ and 1/p + 1/q = 1. If the ideal I has property (AP), then: (i) A : c b∗I −→ c if and only if ∑ k |ank|= OI (1), (13) ∃I - lim n ank (k ∈ N), (14) ∃I - lim n ∑ k ank; E. Kolk / Eur. J. Pure Appl. Math, 8 (2015), 357-367 364 (ii) A : c0 b∗I −→ c if and only if (13) and (14) hold; (iii) A : ℓ1 b∗I −→ c if and only if (14) is satisfied, hn = supk |ank| < ∞ (n ∈ N) and (hn) is I -bounded; (iv) A : ℓp b∗I −→ c if and only if (14) is satisfied and ∑ k |ank| q = OI (1). Letting I = IT in Propositions 2, 3 and Corollary 1, we get the characterizations of ana- logical matrix maps in the sense of b∗T -statistical convergence. We also remark that the matrix maps in the sense of bI - and bstT -convergence were studied in [15]. At the beginning of Section 2 we remarked that a weakly I -convergent sequence is not necessary I -bounded. This fact leads us to a new variant of weak I -convergence. Definition 3. A sequence x = (xn) ∈ ω(X ) is said to be weakly b∗I -convergent to l ∈ X , briefly wb∗I - limn xn = l, if x is weakly I -convergent to l and there is a set K ∈ F (I ) such that the sequence (x ′(xk))k∈K is bounded for every x ′ ∈ X ′. For I = IT we get the notion of weak b∗T-statistical convergence, in this case we write wb∗stT - limn xn = l. Using bounded linear functionals Fz : X ′ → R, Fz x ′ = x ′(z) (x ′ ∈ X ′, z ∈ X ), we can say that wb∗I - limn xn = l (wb∗stT - limn xn = l) if and only if the sequence (Fxn ) is b∗I -convergent (b∗T -statistically convergent) to Fl . Thus, since ‖Fz‖ = ‖z‖, by Theorems 4 and 5 we get the following characterizations of these new types of weak convergence. Proposition 4. Let x = (xn) ∈ ω(X ) and l ∈ X . Assume that X ′ has a countable fundamental set Φ′. (i) If I is an ideal with the property (AP), then wb∗I - limn xn = l if and only if ‖xn‖=OI (1), (15) I - lim n φ′(xn) =φ ′(l) (φ′ ∈ Φ′). (16) (ii) If T is a regular matrix, then wb∗stT - limn xn = l if and only if (15) and (16) are satisfied with stT instead of I . Finally we apply Proposition 4 to Banach sequence spaces c0(X ) and ℓp(X ) with 1 < p <∞. It is known that the dual spaces c0(X ) ′ and ℓp(X ) ′ are isometrically isomorphic, respectively, to ℓ1(X ′) and ℓq(X ′), where 1/p + 1/q = 1 (see, for example, [18]). If Φ′ is a fundamental set of X ′, then E0(Φ ′) is the fundamental set of ℓ1(X ′) and ℓq(X ′). Thus from Proposition 4 we get the following two corollaries. Corollary 2. Let xn = (xni) (n ∈ N) and x0 = (x i) be the elements of c0(X ). Assume that the dual X ′ has a countable fundamental set Φ′. REFERENCES 365 (i) If I is an ideal with the property (AP), then wb∗I - limn xn = x0 if and only if ‖xn‖∞ =OI (1), (17) I - lim i φ′(xni) =φ ′(x i) (φ ′ ∈ Φ′, n ∈ N). (18) (ii) If T is a non-negative regular matrix, then wb∗stT - limn xn = x0 if and only (17) and (18) hold with stT instead of I . Corollary 3. Let xn = (xni) (n ∈ N) and x0 = (x i) be the elements of ℓp(X ) (1 < p <∞). Assume that X ′ has a countable fundamental set Φ′. 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