Banach Algebras of Matrix Transformations Between Spaces of Strongly Bounded and Summable Sequences

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1 Advances in Dynamical Systems and Applications ISSN , Volume 6, Number, pp Banach Algebras of Matrix Transformations Between Spaces of Strongly Bounded and Summable Sequences Eberhard Malowsy Fatih University Department of Mathematics Büyüçemece, Istanbul, Turey emalowsy@fatih.edu.tr eberhard.malowsy@math.uni-giessen.de Abstract Let p <. We characterise the classes X, Y of all infinite matrices that map X into Y for X = w p Λ or X = w p 0 Λ and Y = w Λ, for X = w Λ and Y = w p Λ, and for X = M p Λ and Y = M Λ, the β-duals of w p Λ and w Λ. As special cases, we obtain the characterisations of the classes of all infinite matrices that map w p into w, and w p into w. Furthermore, we prove that the classes w Λ, w Λ and w, w are Banach algebras. AMS Subject Classifications: 46A45, 40H05. Keywords: Spaces of strongly Λ-summable and bounded sequences, matrix transformations, Banach algebras. Introduction Maddox [5] introduced the set w p of all complex sequences x = x that are strongly summable with index p by the Cesàro method of order ; that is, w p contains all sequences x for which lim n σ p nx; ξ = 0 for some complex number ξ, where σ p nx; ξ = n + n x ξ p for all n = 0,,.... We will also consider the sets w p 0 and w p of all sequences that are strongly summable to zero and strongly bounded, with index p; that is, the sets w p 0 and w p contain all Received September 30, 200; Accepted November 22, 200 Communicated by Mališa R. Žižović

2 92 Eberhard Malowsy sequences x for which lim σ p n nx; 0 = 0 and sup σnx; p 0 <, respectively. Maddox n also established necessary and sufficient conditions on the entries of an infinite matrix to map w p into the space c of all convergent sequences; his result is similar to the famous classical result by Silverman Toeplitz which characterises the class c, c of all matrices that map c into c, the so-called conservative matrices. Characterisations of classes of matrix transformations between sequence spaces constitute a wide, interesting and important field in both summability and operator theory. These results are needed to determine the corresponding subclasses of compact matrix operators, for instance in [,3], and more recently, of general linear operators between the respective sequence spaces, for instance in [2,7]. They are also applied in studies on the invertibility of operators and the solvability of infinite systems of linear equations, for instance in [6,8]. To be able to apply methods from the theory of Banach algebras to the solution of those problems, it is essential to determine if a class of linear operators of a sequence space X into itself is a Banach algebra; this is nontrivial if X is a BK space that does not have AK. Finally the characterisations of compact operators can be used to establish sufficient conditions for an operator to be a Fredholm operator, as in [3]. The spaces w Λ p and w0λ p for exponentially bounded sequences Λ and p < were introduced in [0]; they are generalisations of the spaces w p and w0. p Their dual spaces were determined in []. In this paper, we establish the new characterisations of the classes X, Y of all infinite matrices that map X into Y for X = w Λ p or X = w0λ p and Y = w Λ, for X = w Λ and Y = w Λ p, and when X is the β-dual of w Λ p or w0λ p and Y is the β-dual of w Λ. As a special case, we obtain the characterisations of the classes of all infinite matrices that map w p into w, and w p into w, the last result being similar to Maddox s and the Silverman Toeplitz theorems. Furthermore, we prove that the classes w Λ, w Λ and w, w are Banach algebras. Our results would be essential for further research in the areas mentioned above. 2 Notations and Known Results Let ω denote the set of all sequences x = x, and l, c 0 and φ be the sets of all bounded, null and finite complex sequences, respectively; also let cs, bs and { } l p = x ω : x p < for p < be the sets of all convergent, bounded and absolutely p-summable series. We write e and e n n = 0,,... for the sequences with e = 0 for all, and e n n = and e n = 0 for n. A sequence b n in a linear metric space X is called a Schauder basis of X if for every x X there exists a unique sequence λ n of scalars such that x = λ n b n. n

3 Banach Algebras of Matrix Transformations 93 A BK space X is a Banach sequence space with continuous coordinates P n x = x n n N 0 for all x X; a BK space X φ is said to have AK if x = x e for every sequence x = x X. Let X be a subset of ω. Then the set X β = {a ω : ax = a x cs for all x X} is called the β-dual of X. Let A = a n be an infinite matrix of complex numbers and x = x ω. Then we write A n = a n n = 0,,... and A = a n n=0 = 0,,... for the sequences in the n-th row and the -th column of A, and A n x = a n x provided the series converges. Given any subsets X and Y of ω, then X, Y denotes the class of all infinite matrices A that map X into Y, that is, A n X β for all n, and Ax = A n x n=0 Y. Let X and Y be Banach spaces and B X = {x X : x } denote the unit ball in X. Then we write BX, Y for the Banach space of all bounded linear operators L : X Y with the operator norm L = sup Lx. We write X = BX, C x B X for the continuous dual of X with the norm f = sup fx for all f X. The x B X following results and definitions are well nown. Since we will frequently apply them, we state them here for the reader s convenience. Proposition 2.. Let X and Y be BK spaces. a Then we have X, Y BX, Y ; this means that if A X, Y, then L A BX, Y, where L A x = Ax x X see [4, Theorem 4.2.8]. b If X has AK then we have BX, Y X, Y ; this means every L BX, Y is given by a matrix A X, Y such that Lx = Ax x X see [4, Theorem.9]. A nondecreasing sequence Λ = λ n n=0 of positive reals is called exponentially bounded if there is an integer m 2 such that for all nonnegative integers there is at least one term λ n in the interval I m = [m, m + ] [0]. It was shown [0, Lemma ] that a nondecreasing sequence Λ = λ n n=0 is exponentially bounded, if and only if the following condition holds I { There are reals s t such that for some subsequence λn =0 0 < s λ n /λ n+ t < = 0,,...; such a subsequence is called an associated subsequence. Example 2.2. A simple, but important exponentially bounded sequence is the sequence Λ with λ n = n + for n = 0,,...; an associated subsequence is given by λ n = 2, = 0,,....

4 94 Eberhard Malowsy Throughout, let p < and q be the conjugate number of p, that is, q = for p = and q = p/p for < p <. Also let n n=0 be a nondecreasing sequence of positive reals tending to infinity. Furthermore let Λ = λ n n=0 be an exponentially bounded sequence, and λ n =0 an associated subsequence with λ n0 = λ 0. We write K <> = 0,,... for the set of all integers with n n +, and define the sets { } w p 0 = w p = w p 0Λ = x ω : lim n { { x ω : sup n x ω : lim n n n x p = 0 } n x p <, λ n+ K <> x p, = 0 }, and w p Λ = { x ω : sup λ n+ K <> x p < }. If p =, we omit the index p throughout, that is, we write w 0 = w etc., for short. Proposition 2.3 See [0, Theorem a, b]. Let n n=0 be a nondecreasing sequence of positive reals tending to infinity, Λ = λ n n=0 be an exponentially bounded sequence and λ n n=0 be an associated subsequence. a Then w p 0 and w p are BK spaces with the sectional norm defined by and w p 0 has AK. x = sup n n /p n x p b We have w p 0Λ = w p 0Λ, w p Λ = w p Λ, and the sectional norm Λ and the bloc norm Λ with x Λ = sup λ n+ are equivalent on w p 0Λ and on w p Λ. K <> x p /p

5 Banach Algebras of Matrix Transformations 95 Remar 2.4. basis. a It can be shown that w p Λ is not separable, and so has no Schauder b It follows from [4, Corollary 4.2.4] and Proposition 2.3, that w p 0Λ and w p Λ are BK spaces with the norm Λ and that w p 0Λ has AK. Example 2.5. We might also define the set w p Λ = w p 0Λ e = {x ω : x ξ w p 0 for some complex number ξ }. It can be shown that the strong Λ-limit ξ of any x w p Λ is unique if and only if λ = lim sup and that w p Λ w p Λ if and only if Λ = sup n + n λ n+ > 0, n + n λ n+ <. In view of Proposition 2.3 b and Example 2.2, the sets w p 0Λ and w p Λ reduce to the BK spaces w p 0 and w p for λ n = n + for n = 0,,...; it is also clear that then Λ < and λ > 0, and consequently w p is a BK space and the strong limit ξ of each sequence x w p is unique. Throughout, we write = Λ, for short. The β-duals play a much more important role than the continuous duals in the theory of sequence spaces and matrix transformations. Let a be a sequence and X be a normed sequence space. Then we write a X = sup x B X a x provided the expression on the right exists and is finite, which is the case whenever X is a BK space and a X β by [4, Theorem 7.2.9]. If Λ is an exponentially bounded sequence with an associated subsequence λ n, then we write max and for the maximum and sum taen over all K <>. We denote by x <> = x e N 0 the -bloc of the sequence x. Parts a and b of the next result are [, Theorem 5.5 a, b], Part c is [, Theorem 5.7], and Parts d and e are [, Theorem 5.8 a, b]. Proposition 2.6. Suppose Λ = λ n n=0 is an exponentially bounded sequence and let λ n =0 be an associated subsequence. We write { } /p M p Λ = a ω : a MpΛ = λn+ a <> q <. =0

6 96 Eberhard Malowsy a Then we have w p 0Λ β = w p Λ β = M p Λ and MpΛ = w p Λ = w p 0 Λ on M pλ. 2. b The continuous dual w p 0Λ of w p 0Λ is norm isomorphic to M p Λ with the norm MpΛ. c Then M p Λ is a BK space with AK with respect to MpΛ. d We have w p Λ ββ = w p 0Λ ββ = w p Λ and M pλ = on M p Λ β. 2.2 e The continuous dual M p Λ of M p Λ is norm isomorphic to w p Λ. Remar 2.7. a The continuous dual of w Λ is not given by a sequence space. b The set w p Λ is β-perfect, that is, w p Λ ββ = w p Λ. 3 Matrix Transformations on w p Λ and w p 0 Λ Let Λ = λ and Λ = λ m m=0 be exponentially bounded sequences and λ =0 and λ m =0 be associated subsequences. Furthermore, let K <> = 0,,... and M <> = 0,,... be the sets of all integers and m with + and m m m +. If A = a m m, is an infinite matrix and M = M =0 is a sequence of subsets M of M <> for = 0,,..., we write S M A for the matrix with the rows S M A = A m, that is, s M A = a m for all, = 0,,.... m M m M Here we establish necessary and sufficient conditions for an infinite matrix A to be in the classes w p Λ, w Λ, w p 0Λ, w Λ and M p Λ, MΛ, and consider the special cases of w p, w and w p, w. We also estimate the operator norms of L A in these cases. Those characterisations and estimates are needed in the proofs of our results on Banach algebras of matrix transformations. First we characterise the classes w p Λ, w Λ and w p 0Λ, w Λ, and estimate the operator norm of L A when the matrix A is a member of those classes. Theorem 3.. Let Λ = λ and Λ = λ m m=0 be exponentially bounded sequences and λ =0 and λ m =0 be associated subsequences. Then we have A w Λ, p w Λ if and only if A Λ,Λ = sup max S M M M <> A < ; 3. M pλ λ m+

7 Banach Algebras of Matrix Transformations 97 moreover, we have w p Λ, w Λ = w p 0Λ, w Λ. If A w p Λ, w Λ, then the operator norm of L A satisfies A Λ,Λ L A 4 A Λ,Λ. 3.2 Proof. Throughout the proof, we write A = A Λ,Λ, for short. First we assume that the condition in 3. is satisfied. Let m N 0 be given. Then there is a unique m N 0 such that m M <m>. We choose M m = {m}, and it follows from 3. that A m MpΛ <, that is, A m w Λ p β by Proposition 2.6 a. Thus we have shown A m w Λ β for all m N 0. Now let x w Λ p be given. For each N 0, we write M x for a subset of M <> for which A m x m M x = max M M A m x <> m M, and put Mx = M x =0. Then we have by a well-nown inequality see [2], 2. and 3. A λ m x 4 max m + λ m M m + M M A m x <> <> m M = 4 λ m + a m x = 4 λ m + x 4 λ m + 4 λ m + Hence it follows that Ax = sup m M x m M x a m S Mx Ax 4 S Mx λ A m + M x pλ x 4 A x < for all. max M M <> S M A M pλ λ m + m M <> A m x 4 A x <, 3.3 and consequently Ax w Λ for all x w Λ. p Thus, we have shown that if the condition in 3. is satisfied, then A w Λ, p w Λ w0λ, p w Λ. Conversely, we assume A w0λ, p w Λ. Then we have A m w0λ p β for all m N 0, hence A m MpΛ < for all m by Proposition 2.6 a. Since w0λ p and w Λ are a BK spaces by Remar 2.4 b, it follows from Proposition 2. a that L A Bw0Λ, p w Λ, and so L A <. We also have L M w0λ p for all M M <> and all N 0, where L M x = λ m+ A m x for all x m M

8 98 Eberhard Malowsy w p 0Λ. Since trivially L M x L A x L A x for all x w p 0Λ, all M M <> and all N 0, it follows by 2. in Proposition 2.6 a that L M MpΛ = L M w p 0 Λ L A for all M M <> and N 0, and so sup max L MpΛ M M M <> = sup max λ M m+ M <> A m m M MpΛ = A L A <. 3.4 Thus we have shown that if A w p 0Λ, w Λ, then 3.3 is satisfied. It remains to show that if A w p Λ, w Λ, then 3.2 holds. But the first and second inequalities in 3.2 follow from 3.4 and 3.3, respectively. Now we characterise the class M p Λ, MΛ, and estimate the operator norm of L A when A M p Λ, MΛ. We write T for the set of all sequences t = t =0 such that for each = 0,,... there is one and only one t M <>. Theorem 3.2. Let Λ = λ and Λ = λ m m=0 be exponentially bounded sequences and λ =0 and λ m =0 be associated subsequences. Then we have A M p Λ, MΛ if and only if A MpΛ,MΛ = sup sup λ N N 0 t T m+a t <, 3.5 N finite N Λ where, of course, λ m+a t N Λ = sup λ λ m+a t, + K <> N If A M p Λ, MΛ, then the operator norm of L A satisfies p /p. A MpΛ,MΛ L A 4 A MpΛ,MΛ. 3.6 Proof. Throughout the proof, we write A = A MpΛ,MΛ, for short. First we assume that the condition in 3.5 is satisfied. Let m N 0 be given. Then there is a unique m N 0 such that m M <m>. We choose N = {m} and t m = m. Then it follows from 3.5 that A m Λ = sup = λ m m+ λ + /p a m p K <> sup λ m m+a tm <, m N Λ

9 Banach Algebras of Matrix Transformations 99 and so A m w Λ p = M p Λ β by Proposition 2.6 a and d. Now let 0 N 0 and x M p Λ be given. For each N 0 with 0 0, let m; x be the smallest integer in M <> such that max A mx = A m;x x. Then we have by a well-nown m M <> inequality see [2] and 2.2 in Proposition 2.6 d max A 0 mx = λ m+ A m;x x m M <> =0 4 max λ N {0,..., 0 } m+a m;x x N = 4 max λ N {0,..., 0 } m+a m;x, x N p /p 4 max sup λ N {0,..., 0 } λ + m+a m;x, x MpΛ K <> N 4 max λ N {0,..., 0 } m+a m;x x MpΛ N Λ 4 sup sup λ N N 0 t T m+a t x MpΛ = 4 A x MpΛ <. N finite N Λ 0 λ m+ =0 Since 0 N 0 was arbitrary, we obtain Ax MΛ 4 A x MpΛ < for all x M p Λ, 3.7 and consequently Ax MΛ for all x M p Λ. Thus we have shown that if the condition in 3.5 is satisfied, then A M p Λ, MΛ. Conversely, we assume A M p Λ, MΛ. Then A m M p Λ β = w p Λ for all m N 0 by Proposition 2.6 a and d. Furthermore, since M p Λ and MΛ are BK spaces by Proposition 2.6 c, it follows from Proposition 2. a that L A BM p Λ, MΛ. We also have L N,t M p Λ for all finite subsets N of N 0 and all sequences t T, where L N,t x = N λ m+a t x for all x M p Λ. Since trivially L N,t x =0 λ m+ max A mx = Ax MΛ m M <> L A x Λ for all finite subsets N of N 0 and all t T, it follows by 2.2 in Proposition 2.6 c that L N,t M = L pλ N,t Λ = λ m+a t L A <. N Λ

10 00 Eberhard Malowsy Since this holds for all finite subsets N of N 0 and all t T, we conclude A = sup sup λ N N 0 t T m+a t L A <. 3.8 N finite N Λ Thus we have shown that if A M p Λ, MΛ, then 3.5 is satisfied. Finally, if A M p Λ, MΛ, then 3.6 follows from 3.8 and 3.7. Using the transpose A T of a matrix A, we obtain an alternative characterisation of the class w Λ, w p Λ. Theorem 3.3. We have A w Λ, w Λ p if and only if A T MpΛ <. 3.9,MΛ Proof. Since X = w 0 Λ and Z = M p Λ are BK spaces with AK by Remar 2.4 b and Proposition 2.6 c, and Y = Z β = w p Λ by Proposition 2.6 d, it follows from [4, Theorem 8.3.9] that A w 0 Λ, w p Λ = X, Y = X ββ, Y = w Λ, w p Λ and A w 0 Λ, w p Λ if and only if A T Z, X β = M p Λ, MΛ, and, by 3.5 in Theorem 3.2, this is the case if and only if 3.9 holds. We consider an application to the characterisations of the classes w p, w, w p, w and w, w p. Let Λ = Λ and λ n = n + for n = 0,,... as in Examples 2.2 and 2.5. Then we may choose the subsequences given by λ = 2 and λ m = 2 for all, = 0,,..., and consequently the sets K <> and M <> are the sets of all integers and m with and 2 m 2 +. We also write M p = M p Λ. Remar 3.4. a We obviously have w p 0 w p w p. b For each x w p, the strong limit ξ, that is, the complex number ξ with lim x 2 ξ p = 0 is unique see [5]. c Every sequence x = x w p has a unique representation x = ξ e + x ξe [5]. Example 3.5. a It follows from Theorem 3. that A w, p w = w0, p w if and only if A w p,w = sup max 2 M M <, 3.0 <mu> A m m M Mp

11 Banach Algebras of Matrix Transformations 0 where 2 max K a m <> =0 m M A m = m M Mp 2 /p =0 K <> m M a m q /q p = < p <. b It follows from Part a and [4, 8.3.6, 8.3.7] that A w p, w if and only if 3.0, for each there exists a complex number α with lim a 2 m α = 0 3. m M <> and hold. lim 2 m M <> a m α = 0 for some comlex number α 3.2 c We obtain from Theorems 3.2 and 3.3, interchanging the roles of N and K, and and, that A w, w p if and only if sup K N 0 K finite sup t T 2 A t <, K Λ where 2 A t K Λ = sup 2 m M <> 2 a m,t. K We also give a formula for the strong limit of Ax when A w p, w and x w p. Theorem 3.6. If A w p, w, then the strong limit η of Ax for each sequence x w p is given by η = α ξ + α x ξ, 3.3 where ξ is the strong limit of the sequence x, and the complex numbers α and α for = 0,,... are given by 3.2 and 3. in Example 3.5 b.

12 02 Eberhard Malowsy Proof. We assume A w p, w and write = w p,w, for short. The complex numbers α and α for = 0,,... exist by Example 3.5 b. First, we show α M p. Let x w p and 0 N 0 be given. Then there exists an integer 0 with 0 K <0> and we have by the inequality in [9, Lemma ] 0 0 α x = α 2 x 0 a 2 n α x + 0 a 2 n α 0 a 2 n α 0 =0 x + 4 max M M <> x + 4 sup a 2 n x 2 2 =0 max M M <> n M a n A n n M Mp x x. 0 Letting tend to, we obtain α x A < from 3. and 3.0. Since 0 N 0 was arbitrary, it follows that α = w p β = M p. Now we write ˆαx = α x < for all x w p, that is, α x and B = b n n, for the matrix with b n = a n α for all n and, and show lim B 2 n x = 0 for all x w0. p 3.4 Let x w p 0 and ε > 0 be given. Since w p 0 has AK, there is 0 N 0 such that x x [ 0 ] < ε A + α M p + for x[ 0] = 0 x e. It also follows from 3. that there is 0 N 0 such that B 2 n x [ 0] = 0 b 2 n x < ε for all 0. Let 0 be given. Then we have B 2 n x Bn x [ 0] + Bn x x [ 0 ] 2 2

13 Banach Algebras of Matrix Transformations 03 < ε + 4 max ε + 4 M M <> max M M <> B 2 n x x [0] n M 2 x x [ 0 ] < 5 ε. B n n M Mp Thus we have shown 3.4. Finally, let x w p be given. Then there is a unique complex number ξ such that x 0 = x ξ e w0, p by Remar 3.4 b, and we obtain by 3.4 and A 2 n x η = 2 A nx 0 + ξ A n e α ξ + α x 0 An x 0 ˆαx 0 + ξ A 2 2 n e α = B 2 n x 0 + ξ a 2 n α = 0. This completes the proof. 4 The Banach Algebra w Λ, w Λ In this section, we show that w Λ, w Λ is a Banach algebra with respect to the norm defined by A = L A for all A w Λ, w Λ. We also consider the nontrivial special case of w, w. We need the following results. Lemma 4.. in fact a The matrix product B A is defined for all A, B w Λ, w Λ; b nm a m B n MΛ A for all n and. 4. m=0 b Matrix multiplication is associative in w Λ, w Λ. c The space w Λ, w Λ is a Banach space with respect to A Λ,Λ = sup max λ + max λ m+ M M <> K <> =0 m M a m. 4.2

14 04 Eberhard Malowsy Proof. a Let A, B w Λ, w Λ. First we observe that e w Λ implies Ae = A m e m=0 = a m m=0 = A w Λ for all. Therefore we have A = sup a m < for all. 4.3 λ m+ m M <> Furthermore B w Λ, w Λ implies B n w Λ β = MΛ for all n, that is, by Proposition 2.6 a B n MΛ = λ m+ max b nm < for all n. 4.4 m M <> =0 Now it follows from 4.3 and 4.4 that Bn A b nm a m = λ m+ b nm a m λ m=0 =0 m+ m M [ <> ] λ m+ max b nm a m m M <> λ =0 m+ m M <> λ m+ max b nm sup a m m M <> λ =0 m+ m M <> = B n MΛ A < for all n and. b Let A, B, C w Λ, w Λ. We write for D w Λ, w Λ M T D = D T MΛ,MΛ = sup K N 0 K finite sup t T sup λ m+ m M <> K λ n+ d m,t and note that M T D < by Theorem 3.3. We are going to show that the series a nm b m c j are absolutely convergent for all n and j. We fix n and j and write m=0 s = A n and t = C j for the sequences in the n-th row of A and the j-th column of C. Then we have s MΛ and t w Λ. We define the matrix D = d, by d = b m for, = 0,,.... λ m+ m M <> Furthermore, given N 0, for every = 0,,..., let = K <> be the smallest integer with max d K <> = d. Then by the inequality in [9, Lemma ], λ m+ D MΛ = λ + b m =0 m M <>

15 Banach Algebras of Matrix Transformations sup K N 0 K finite sup K N 0 K finite max M M λ + <> K λ + b m, m M <> K m M b m hence It also follows that for = 0,,... D MΛ 4 M T B < for = 0,, λ m+ m M <> b m t = t d D MΛ t. 4.6 Therefore, we obtain from 4.6 and 4.5 m=0 s m b m t sup Thus we have shown that sup b m t s MΛ λ m+ m M <> D MΛ t s MΛ 4 M T B t s MΛ <. m=0 s m b m t is absolutely convergent, and consequently matrix multiplication is associative in w Λ, w Λ. c We assume that A j j=0 is a Cauchy sequence in w Λ, w Λ. Since w Λ, w Λ = w 0 Λ, w Λ by Theorem 3. and w 0 Λ has AK by Remar 2.4 b, it is a Cauchy sequence in w 0 Λ, w Λ = Bw 0 Λ, w Λ, by Proposition 2.. Consequently there is L A Bw 0 Λ, w Λ with L A j L A. Since w 0 Λ has AK there is a matrix A w 0 Λ, w Λ by Proposition 2. b such that Ax = L A x for all x w 0 Λ. Finally w 0 Λ, w Λ = w Λ, w Λ implies A w Λ, w Λ. The following result is obtained as an immediate consequence of Lemma 4.. Theorem 4.2. The class w Λ, w Λ is a Banach algebra with respect to the norm A = L A for all A w Λ, w Λ. The following example is obtained from Theorem 4.2. Example 4.3. Let λ n = n + for n = 0,,... as in Examples 2.2, 2.5 and 3.5. Then w, w is a Banach algebra with A = L A. Finally, we show that w, w is a Banach algebra.

16 06 Eberhard Malowsy Theorem 4.4. The class w, w is a Banach algebra with A = L A. Proof. We have to show in view of Theorem 4.2 that i w, w is complete; ii if A, B w, w, then B A w, w. First we show i. Let A j j= be a Cauchy sequence in w, w. Since w, w w, w and the operator norm on Bw, w is the same as that on Bw, w, it follows that A j j= is a Cauchy sequence in w, w, and so A = lim A j j w, w by Lemma 4. c. We have to show A w, w. Let ε > 0 be given. Since A j j= is a Cauchy sequence in w, w there exists a j 0 N 0 such that A j A l w,w = sup 2 max M M <> A j n n M A l n M < ε 4 for all j, l j 0; 4.7 Also, by 3. and 3.2, for each fixed j there exist complex numbers α j = 0,,... and α j such that and lim lim 2 a j n αj 2 a j n αj Let j, l j 0 be given. Then we have for each fixed N 0 by 4.7 α j α l = α j α l a j n αj a j n αj + 4 max M M <> = 0 for each 4.8 = a l n αl a l n αl A j 2 n n M + 2 A l n e a j n al n

17 Banach Algebras of Matrix Transformations a j n αj + 2 a l n αl + 4 sup max A j A l 2 M M <> M a j n αj + 2 a l n αl Letting, we obtain from 4.8 α j α l ε for all j, l j 0. e + ε for all N 0. Thus α j j= is a Cauchy sequence of complex numbers for each fixed N 0 and so α = lim α j j exists for each N Now let N 0 be fixed. Then we obtain for all sufficiently large j and for all by 4.0 and since A = lim j A j a 2 n α a j 2 n a n + 2 A j A + w,w 2 < 2 ε + 2 a j n αj. a j n αj a j n αj Letting, we obtain from 4.8 lim a 2 n α ε. + α 2 α j + ε Since ε > 0 was arbitrary, it follows that α satisfies the condition in 3. of Example 3.5 b. Using exactly the same argument as before with a j n and αj replaced by a j n and αj, and applying 4.9 instead of 4.8, we conclude that α = lim α j j exists and satisfies the condition in 3.2 of Example 3.5 b. Finally A w, w and 3. and 3.2 imply A w, w by Example 3.5 b. Thus we have shown that w, w is complete. This completes the proof of i. Now we show that A, B w, w implies B A w, w. Since A, B w, w, by Example 3.5 b, there are complex numbers α, α that satisfy 3. and 3.2, and

18 08 Eberhard Malowsy complex numbers β, β that satisfy 3. and 3.2 with b n, β and β instead of a n, α and α. Let x w be given and ξ be the strong limit of x. We put ζ = β β n α α ξ + α x + β n A n x. n=0 We observe that α, β n n=0 M by the proof of Theorem 3.6, and also trivially M l cs. Therefore all the series in the definition of ζ converge. We write C = B A, y = Ax, η for the strong limit of the sequence y, and ζ for the strong limit of the sequence z = By. Since Cx = BAx by Lemma 4. b, we obtain by 3.3 in Theorem 3.6 C m x ζ = B m y ζ = B my β n y n β β n α α ξ + n=0 n=0 = B my β n y n β β n η n=0 n=0 = z β m β n y n η + η = z m ζ for all m, n=0 n=0 α x hence lim C 2 m x ζ m M <> = lim z 2 m ζ m M <> = 0. This shows that Cx w, and completes the proof of ii. References [] L. W. Cohen, N. Dunford, Transformations on sequence spaces, Due Mathematical Journal, 3 no. 4: , 937. [2] I. Djolović, E. Malowsy, A note on compact operators on matrix domains, J. Math. Anal. Appl., 73-2:93 23, [3] I. Djolović, E. Malowsy, A note on Fredholm operators on c 0 T Applied Mathematics Letters, 22: , [4] A. M. Jarrah, E. Malowsy, Ordinary, absolute and strong summability and matrix transformations, Filomat, 7:59 78, 2003.

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