Characterization of Some Classes of Compact Operators between Certain Matrix Domains of Triangles

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1 Filomat 30:5 (2016), DOI /FIL D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Characterization of Some Classes of Compact Operators between Certain Matrix Domains of Triangles Ivana Djolović a, Eberhard Malowsy b a Technical Faculty in Bor, University of Belgrade, VJ 12, Bor, Serbia b Department of Mathematics, Faculty of Science, Fatih University, Büyüçemece 34500, Istanbul, Turey c/o Državni Univerzitet u Novom Pazaru, Vua Karadžića bb, Novi Pazar, Serbia Abstract. In this paper, we characterize the classes ((l 1 ) T, (l 1 ) T) and (c T, c T) where T (t n ) and n,0 T ( t n ) are arbitrary triangles. We establish identities or estimates for the Hausdorff measure of n,0 noncompactness of operators given by matrices in the classes ((l 1 ) T, (l 1 ) T) and (c T, c T). Furthermore we give sufficient conditions for such matrix operators to be Fredholm operators on (l 1 ) T and c T. As an application of our results, we consider the class (bv, bv) and the corresponding classes of matrix operators. Our results are complementary to those in [2] and some of them are generalization for those in [3]. 1. Introduction and Notation As usual, let ω, φ, c and c 0 denote the sets of all complex, finite, convergent and null sequences x (x ) 0, respectively, and l 1 {x ω : 0 x < } be the set of all absolutely convergent series. We write e and e (n) (n 0, 1,... ) for the sequences with e 1 for all, and e (n) n 1 and e (n) 0 ( n), respectively. The β dual of a subset X of ω is the set X β {a ω : 0 a x converges for all x X}. Let A (a n ) be an infinite matrix of complex numbers, X and Y be subsets of ω and x ω. We write n,0 A n (a n ) 0 for the sequence in the n th row of A, A nx 0 a nx and Ax (A n x) (provided all the series A n x converge). The set X A {x ω : Ax X} is called the matrix domain of A in X. Also (X, Y) is the class of all matrices A such that X Y A ; so A (X, Y) if and only if A n X β for all n and Ax Y for all x X. A Banach space X ω is a BK space if each projection x x n on the n th coordinate is continuous. A BK space X φ is said to have AK if x [m] m 0 x e () x (m ) for every sequence x (x ) 0 X. If X φ is a BK space and a ω we write a X a x : x 1 0 provided the expression on the righthand side is defined and finite which is the case whenever a X β ([12, Theorem 7.2.9]) Mathematics Subject Classification. Primary 46B45 ; Secondary 47B37 Keywords. Sequence spaces, matrix domains of triangles, matrix transformations, Hausdorff measure of noncompactness, compact operators, Fredholm operators Received: 24 November 2014; Accepted: 16 January 2015 Communicated by Dragan S. Djordjević Research of the authors ported by the research projects # and #174025, respectively, of the Serbian Ministry of Science, Technology and Environmental Development, and of the second author also by the project #114F104 of Tubita addresses: zuco@open.teleom.rs (Ivana Djolović), eberhard.malowsy@math.uni-giessen.de (Eberhard Malowsy)

2 I. Djolović, E. Malowsy / Filomat 30:5 (2016), If X and Y are Banach spaces, then we write, as usual, B(X, Y) for the set of all bounded linear operators L : X Y with the operator norm defined by L x 1 { L(x) }. The following result is very important for our research. Lemma 1.1. Let X and Y be BK spaces. (a) Then we have (X, Y) B(X, Y), that is, every A (X, Y) defines an operator L A B(X, Y) where L A (x) Ax for all x X ([5, Theorem 1.23] or [12, Theorem 4.2.8]). (b) If X has AK then we have B(X, Y) (X, Y), that is, every L B(X, Y) is given by a matrix A (X, Y) such that Ax L(x) for all x X ([4, Theorem 1.9.]). In this paper we consider the classes ((l 1 ) T, (l 1 ) T) and (c T, c T) where T (t n ) n,0 and T ( t n ) n,0 are triangles. A matrix T (t n ) n,0 is said to be a triangle if t n 0 for all > n and t nn 0 (n 0, 1...). Throughout, let T denote a triangle, S its inverse and R S t, the transpose of S. We remar that the inverse of a triangle exists, is unique and a triangle ([12, 1.4.8, p. 9] and [9, Remar 22 (a), p. 22]). In [2], the authors considered the space (c 0 ) T {x ω : Tx c 0 }, generalized some results on matrix transformations and compact operators on (c 0 ) T, and finally gave a sufficient condition for a linear operator on (c 0 ) T defined by an infinite matrix to be a Fredholm operator. Since many recently defined sequence spaces arise from the concept of matrix domains of triangles in classical sequence spaces, we classify the following classes: ((c 0 ) T, (c 0 ) T), (c T, c T), ((l 1 ) T, (l 1 ) T). The class ((c 0 ) T, (c 0 ) T) was the subject of research in [2]. Hence, it remains to consider the classes (c T, c T) and ((l 1 ) T, (l 1 ) T). In this way we extend existing results. Our results are complementary to those in [2] and some of them are generalization for those in [3]. As in [2] and [3], this will be achieved in three steps. First we will characterize the classes (c T, c T) and ((l 1 ) T, (l 1 ) T), then establish identities or inequalities for the Hausdorff measure of noncompactness of the corresponding matrix operators, and give necessary and sufficient conditions for these operators to be compact, and finally, establish sufficient conditions for an infinite matrix to be a Fredholm operator on c T and (l 1 ) T. 2. Matrix Transformations In this section, we characterize the classes (c T, c T) and ((l 1 ) T, (l 1 ) T). These characterizations will be reduced to the well nown characterizations of matrix transformations between the classical sequence spaces ([11, 12]). The following results play an important role in our research. Lemma 2.1. ([5, Theorem 1.23]) Let X be a BK space and A (X, l ). Then we have L A A n. n Lemma 2.2. ([5, Theorem 3.8]) Let T be triangle. (a) Then, for arbitrary subsets X and Y of ω, A (X, Y T ) if and only if B TA (X, Y). (b) If X and Y are BK spaces and A (X, Y T ), then L A L B. Now we establish a slightly improved version of [6, Theorem 3.4]. Lemma 2.3. Let X be a BK space with AK, Y be an arbitrary subset of ω and R S t. Then A (X T, Y) if and only if  (X, Y) and W (A n) (X, l ) for all n 0, 1,..., where  is the matrix with the rows  n RA n for (n 0, 1,... ) and the triangles W (A n) (n 0, 1,... ) are defined by w (A n) m a nj s j (0 m) jm 0 ( > m) (m 0, 1,... ). (2.1)

3 Moreover, if A (X T, Y) then we have I. Djolović, E. Malowsy / Filomat 30:5 (2016), Az Â(Tz) for all z Z X T. (2.2) Proof. First we assume A (X T, Y). Then it follows from [6, Theorem 3.4] that  (X, Y) and W (An) (X, c 0 ) for all N, and so W (An) (X, l ). Conversely we assume that  (X, Y) and W (An) (X, l ) for all n. Then the series â n j a njs j converge for each and n, and so jm a nj s j w (A n) m 0 for each n and each fixed, and this and W (A n) (X, l ) together imply W (A n) (X, c 0 ) by [12, 8.3.6], since X is an FK space with AK and c 0 is a closed subspace of l. Now it follows from [6, Theorem 3.4] that A (X T, Y). The last part is now obvious. We also need the next result. Lemma 2.4. ([6, Theorem 3.6]) Let X and Y be BK spaces and X have AK. If A (X T, Y) then we have L A L (2.3) where  is the matrix defined in Lemma 2.3. Before characterizing the class ((l 1 ) T, (l 1 ) T), it is useful to observe the following. Remar 2.5. Let X be a BK space with AK and Y be an arbitrary subset of ω. Applying Lemma 2.2 (a) first and then Lemma 2.3, we obtain A (X T, X T) in and only if ˆB (X, Y) and W (Bn) (2.4) (X, l ) for all n, where B TA, ˆb n j s jb nj for all n and, and m s j b nj (0 m) jm 0 ( > m) (m 0, 1,... ). On the other hand, applying Lemma 2.3 first and then Lemma 2.2 (a), we obtain A (X T, Y T) if and only if Ĉ T (X, Y) and W (An) (X, l ) for all n, (2.5) Then the conditions in (2.4) and (2.5) are equivalent. Proof. First we assume that the conditions in (2.4) are satisfied. Then the series ˆb n s j b nj j j s j t ni a ij converge for all n and. (2.6) i0 If we fix then it is easy to show by mathematical induction with respect to n that the series â n s j a nj converge for each n 0. (2.7) j

4 I. Djolović, E. Malowsy / Filomat 30:5 (2016), It follows from (2.7) that ˆb n j s j t ni a ij i0 i0 t ni s j a ij j t ni â i ĉ n for all n and, i0 and consequently the condition ˆB (X, Y) implies Ĉ (X, Y). Similarly it can be shown that the convergence of m implies that of w (A n) m s j b nj for all n, 0 m and all m jm s j a nj for all n, 0 m and all m jm and that m s j b nj jm jm s j t ni a ij i0 t ni i0 jm s j a ij i0 t ni w (A i) m. (2.8) Writing S for the inverse of the triangle T we obtain for all n, m and with 0 m s nl w (B l) m l0 l0 s nl l i0 t li w (A i) m i0 w (A i) m s nl t li li i0 w (A i) m δ ni w (A n) m, and so W (B n) (X, l ) for all n implies W (A n) (X, l ) for all n. Thus we have established that the conditions in (2.4) imply those in (2.5). Conversely we assume that the conditions in (2.5) are satisfied. Then the series in (2.7) converge for all n and, hence ˆb n s j b nj j j s j t ni a ij i0 i0 t ni s j a ij j t ni â i ĉ n. (2.9) i0 Thus Ĉ (X, Y) implies ˆB (X, Y). Similarly it can be shown that (2.8) holds for all n, m and 0 m, and consequently A n (X, l ) for all n implies B (n) (X, l ) for all n. Now we prove our first main result. Theorem 2.6. Let T and T be triangles. Then we have A ((l 1 ) T, (l 1 ) T) if and only if s j t ni a ij <, (2.10) j i0 and m, s j a nj < for all n 0, 1,... (2.11) jm

5 I. Djolović, E. Malowsy / Filomat 30:5 (2016), Proof. Since l 1 is a BK space with AK, it follows from Lemma 2.3 that A ((l 1 ) T, (l 1 ) T) if and only if  (l 1, (l 1 ) T) and W (A n) (l 1, l ) for all n. First, we have by Lemma 2.2 that  (l 1, (l 1 ) T) if and only if B T (l 1, l 1 ) which is the case if and only if ([12, 8.4.1D]) b n <. (2.12) It follows from the definition of the matrices B and  that b n t ni â i s j a ij t ni a ij, i0 i0 t ni j j s j i0 and so the conditions in (2.12) and (2.10) are the same. Furthermore, we have W (An) (l 1, l ) by [12, 8.4.1A] if and only if w (A n) m a nj s j <, m, m, which is is (2.11). jm The space c of convergent sequences is not an AK space, so we cannot apply Lemma 2.3 in the characterization of the class (c T, c T). This is why we need the next result. Lemma 2.7. ([6, Remar 3.5 (b)]) Let Y be a linear subspace of ω. Then we have A (c T, Y) if and only if and  (c 0, Y), W (A n) (c, c) for all n (2.13) Âe (α n ) Y where α n ω n (A) 0 w (A n) m for n 0, 1,.... (2.14) Remar 2.8. Let Y be an arbitrary subspace of ω. Applying Lemma 2.2 (a) first and then Lemma 2.7, we obtain A (c T, c T) if and only if ˆB (c 0, c) W (Bn) (c, c) ˆBe (β n ) c for all n where β n ω n (B) 0 m for n 0, 1,.... On the other hand, applying Lemma 2.7 first and then Lemma 2.2 (a), we obtain A (c T, c T) if and only if Ĉ T (c 0, c) W (An) (c, c) for all n T ( Âe (α n ) ) c with αn from (2.14) for n 0, 1,.... Then conditions in (2.15) and (2.16) are equivalent. (2.15) (2.16) Proof. First we assume that the conditions in (2.15) are satisfied. Then it follows as in Remar 2.5 that the first two conditions in (2.16) are satisfied and ĉ n ˆb n for all n and. Furthermore Ĉ (c 0, c) (c 0, l ) (l, l ), that is,  (l, (l ) T), implies e ωã, and so by [12, Theorem (i)] Ĉe ( TA)e T(Ae).

6 I. Djolović, E. Malowsy / Filomat 30:5 (2016), Since W (A n) (c, c) for all n, the its α n exist for all n, and so we obtain by (2.8) for all n ( ) T n (αj ) j0 t ni α i i0 0 i0 t ni m β n. 0 w (A i) m Consequently it follows from the third condition in (2.15) that T(Âe (α n )) T(Âe) T(α n ) Ĉe (β n ) ˆBe (β n ) c. 0 i0 t ni w (A i) m Thus we have shown that the conditions in (2.15) imply those in (2.16). Conversely we assume that the conditions in (2.16) are satisfied. Then it follows as above, that the first two conditions in (2.15) are satisfied, and ˆb n ĉ for all n and. Also ˆB (c 0, c) implies ˆB (l, l ) and so e ω ˆB, and W(B n) (c, c) for all n implies that the its β n exist for all n. Again we have β n T n ((α j ) j0 ) for all n and ˆBe (β n ) T(Âe (α n )), and the third condition in (2.16) implies the third condition in (2.15). Now, we can prove our results of the next theorem. Theorem 2.9. Let T and T be triangles. Then we have A (c T, c T) if and only if the following conditions hold: n ˆbn <, 0 (2.17) ˆb n ˆβ exists for all, m m 0 0 (2.18) < for each n, (2.19) m β n exists for each n, (2.20) ˆb n β n η exists. (2.21) 0 where the matrices B, ˆB and W (B n) are defined as in Remar 2.5. Proof. Applying Lemma 2.2 (a) we have A (c T, c T) if and only if B TA (c T, c), which by Lemma 2.7 is equivalent to ˆB (c 0, c), W (B n) (c, c) for all n and ˆBe ( β n ) c. Furthermore, we have ˆB (c 0, c) if and only if ([12, 8.4.5A]) n 0 ˆb n < and ˆbn ˆβ for all 0, 1,..., (2.22) that is, (2.17) and (2.18). Also ˆBe (β n ) c is the condition in (2.21). Furthermore, we have by [12, 8.4.5A] that ˆB (c 0, c) if and only if the conditions in (2.17) and (2.18) hold. Finally we have W (Bn) (c, c) by [12, 8.4.5A] if and only if the conditions in (2.19) and (2.20) hold and exists for each, the last m condition obviously being redundant.

7 I. Djolović, E. Malowsy / Filomat 30:5 (2016), Compact Operators We recall some definitions and results which are important for this section. If X and Y are Banach spaces then a linear operator L : X Y is said to be compact if its domain is all of X and for every bounded sequence (x n ) in X, the sequence (L(x n)) has a convergent subsequence in Y. We denote the class of such operators by C(X, Y). If X Y, we write C(X), for short. The most effective way to find conditions for a linear operator L to be compact is by applying the Hausdorff measure of noncompactness. Let (X, d) be a metric space, M X denote the class of bounded subsets of X and B(x, r) {y X : d(x, y) < r} be the open ball of radius r > 0 with its centre in x. Then the Hausdorff measure of noncompactness of the set Q M X, denoted by χ(q), is given by χ(q) inf{ɛ > 0 : Q n B(x i, r i ), x i X, r i < ɛ (i 1,..., n), n IN}; i1 the function χ is called the Hausdorff measure of noncompactness. Let X and Y be Banach spaces and χ 1 and χ 2 be Hausdorff measures of noncompactness on X and Y. Then the operator L : X Y is called (χ 1, χ 2 )-bounded if L(Q) M Y for every Q M X and there exists a positive constant C such that χ 2 (L(Q)) Cχ 1 (Q) for every Q M X. If an operator L is (χ 1, χ 2 ) bounded then the number L (χ1,χ 2 ) inf{c > 0 : χ 2 (L(Q)) Cχ 1 (Q) for all Q M X } is called the (χ 1, χ 2 )- measure of noncompactness of L. In particular, if χ 1 χ 2 χ, then we write L (χ,χ) L χ. We need the following results. Lemma 3.1. Let X and Y be Banach spaces and L B(X, Y), S X {x X : x 1} and B X {x X : x 1}. Then we have L χ χ(l( B X )) χ(l(s X )) ([5, Theorem 2.25]); (3.1) L C(X, Y) if anf only if L χ 0 ([5, Corollary 2.26 (2.58)]); (3.2) Lemma 3.2 (Goldenštein, Gohberg, Marus). ([5, Theorem 2.23]) Let X be a Banach space with a Schauder basis (b n ), Q M X and P n : X X be the projector onto the linear span of {b 1, b 2,..., b n }. Then we have 1 a (I P n )(x) χ(q) (I P n )(x), (3.3) x Q x Q where a I P n. (Let us mention that if X c then a 2). Lemma 3.3. ([8, Theorem 2.8.]) Let Q M X where X is l p for 1 p < or c 0. If P n : X X is the operator defined by P n (x) x [m] for all x (x ) X, then 0 χ(q) ( (I P n )(x) ). x Q Lemma 3.4. ([6, Theorem 4.2]) Let X be a normed sequence space and χ T and χ denote the Hausdorff measures of noncompactness on M XT and M X. Then we have χ XT (Q) χ(t(q)) for all Q M XT. Lemma 3.5. Let X and Y be Banach sequence spaces, T be a triangle and L B(X, Y T). Then we have L (χ,χ T ) L T L χ. Proof. We have by (3.1) and Lemma 3.4 L (χ,χ T ) χ T(L(S X )) χ ( T(L(S X )) ) χ ( (L T L)(S X ) ) L T L χ.

8 I. Djolović, E. Malowsy / Filomat 30:5 (2016), Now we establish an identity for the Hausdorff measure of noncompactness of matrix operators in B((l 1 ) T, (l 1 ) T) and necessary and sufficient conditions for such operators to be compact. Theorem 3.6. Let T and T be triangles and the operator L A B((l 1 ) T, (l 1 ) T) be given by a matrix A ((l 1 ) T, (l 1 ) T). Then we have L A (χt,χ T ) ˆb n where ˆb n s j t ni a ij for all n and. j i0 nr Furthermore, L A is compact if and only if s j t ni a ij 0. nr j i0 Proof. Applying Lemma 3.5, [1, 1, Corollary 3.6 (b), (3.14)] and (2.9), we obtain with B TA L A (χt,χ T ) L T L A (χt,χ) L B (χt,χ) ˆb n, that is, the first identity of the theorem. The characterization of compact matrix operators now follows by (3.2). Now we establish an inequality for the Hausdorff measure of noncompactness of matrix operators in B(c T, c T ) and necessary and sufficient conditions for such operators to be compact. Theorem 3.7. Let T and T be triangles and the operator L A B(c T, c T) be given by a matrix A (c T, c T). Then we have where 1 2 M L A χ M M n>r ˆbn ˆβ + ˆβ η β n 0 0 and ˆb n, β, η and ˆβ are defined in Theorem 2.9. L A is compact if and only if ˆbn ˆβ + ˆβ η β n 0. n>r 0 0 Proof. We put for arbitrary matrices D φ n (D) d ˆ n ˆδ + ˆδ η(d) δ n for n 0, 1,..., where 0 0 ˆδ dn ˆ for 0, 1,..., δ n ω n (D) 0 w (D n) m for n 0, 1,... nr

9 I. Djolović, E. Malowsy / Filomat 30:5 (2016), and η(d) ˆ d n δ n. 0 We obtain from Lemma 3.5 and [1, 1, Corollary 3.6 (b), (3.16) (3.19)] that if A (c T, c T) then ( ) ( ) 1 2 φ n (C) L χ φ n (C), where C TA. r n r n We have ĉ n ˆb n for all n and by (2.9) and so ˆγ ĉ n ˆbn ˆβ for 0, 1,... ((2.18) in Theorem 2.9). Furthermore, by definition, m s j b nj jm jm s j t ni a ij i0 jm s j c nj w (C n) m for all n, m and ; this implies γ n 0 w (C n) m 0 m β n ((2.20)) for all n, and finally η(c) ĉ n γ n ˆb n β n η ((2.21)). Thus we have ( M 0 φ n (C) r n ), and the statements of the theorem are an immediate consequence Fredholm Operators on c T and (l 1 ) T Now we establish sufficient conditions for an operator in B(X) B(X, X) to be a Fredholm operator when X l 1 or X c. We recall the definition of a Fredholm operator. Definition 4.1. Let X and Y be Banach spaces and L B(X, Y). We denote the null and the range spaces of L by N(L) and R(L). Then T is said to be a Fredholm operator if the following conditions hold: (1) N(L) is finite dimensional; (2) R(L) is closed; (3) Y/R(L) is finite dimensional. The set of Fredholm operators from X to Y is denoted by Φ(X, Y) and we write Φ(X) Φ(X, X), for short. The next result ([10], p.106) is of greater importance for our studies than the definition itself: if X Y and L C(X), then I L is a Fredholm operator where I is the identity operator on X. We establish sufficient conditions for an operator L A given by a matrix A (X T, X T ) to be a Fredholm operator when X l 1 or X c. Again we use the Hausdorff measure of noncompactness.

10 I. Djolović, E. Malowsy / Filomat 30:5 (2016), We consider infinite matrix C (c n ) n,0 associated with infinite matrix A (a n) and defined as n,0 follows: a n (n ) c n (4.1) 1 a nn (n ). Then we have that if the operator L C given by the infinite matrix C is compact then the operator L A given by the infinite matrix A is a Fredholm operator. Taing this into account, we obtain the following new results: Theorem 4.2. (a) Let L A B((l 1 ) T ) be given by a matrix A. We write D TC and d ˆ n j s jd nj for all n and. If d ˆ n s j t nj t ni a ij 0, nr then we have L A Φ((l 1 ) T ). (b) Let L A B(c T ) be given by a matrix A and D TC. If φ n (D) dn ˆ ˆδ + dˆ n η(d) δ n 0, n r n>r 0 nr j 0 then L A Φ(c T ), where ˆδ ( 0, 1,... ), δ n (n 0, 1,... ) and η(d) are defined as in the proof of Theorem 3.7. Proof. Defining the matrix C as in (4.1), we obtain i0 t ni c ij t nj (1 a jj ) t ni a ij t nj i0 i0,i j t ni a ij. i0 Now, if we apply Theorem 3.6 in (a) and Theorem 3.7 in (b), the proof is obvious since the operator L A given by the infinite matrix A is Fredholm if the operator L C given by the infinite matrix C is compact. We close with an application of our results. Example 4.3. We write bv {x ω : 0 x x 1 < } for the set of all sequences of bounded variation. Let T T be the matrix of the operator of the first difference, that is, nn 1, n,n 1 1 and n 0 for n, n 1 (n 0, 1,... ). Then we have S Σ where Σ n 1 for 0 n and Σ n 0 for > n (n 0, 1,... ), and bv (l 1 ). Now, applying Theorem 2.6 we obtain A (bv, bv) if and only if (a nj a n 1,j ) < ([11, 99.(99.2)]), j since the condition in (2.11) becomes redundant in this case. Also applying Theorem 3.6, we obtain that the matrix operator L A B((l 1 ) T ) is compact if and only if (a nj a n 1,j ) 0. nr j Finally, applying and Theorem 4.2 (b), we obtain that L A Φ((l 1 ) T ) if ( nj (a nj a n 1,j ) ) 0. nr j

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