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Banach algebras of matrix transformations between spaces of strongly bounded and summable sequences.

1 Introduction

Maddox [5] introduced the set [w.sup.p] of all complex sequences x = [([x.sub.k]).sup.[infinity].sub.k=0] that are strongly summable with index p by the Cesaro method of order 1; that is, [w.sup.p] contains all sequences x for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some complex number [xi], where

[[sigma].sup.p.sub.n](x;[xi]) = 1/[n + 1] [n.summation over (k=0)] [[absolute value of [x.sub.k] - [xi]].sup.p] for all n = 0,1,....

We will also consider the sets [w.sup.p.sub.0] and [w.sup.p.sub.[infinity]] all sequences that are strongly summable to zero and strongly bounded, with index p; that is, the sets [w.sup.p.sub.0] and [w.sup.p.sub.[infinity]] contain all sequences x for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Maddox also established necessary and sufficient conditions on the entries of an infinite matrix to map [w.sup.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 [1,13], 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.sup.p.sub.[infinity]]([LAMBDA]) and [w.sup.p.sub.0]([LAMBDA]) for exponentially bounded sequences [LAMBDA] and 1 [less than or equal to] p < [infinity] were introduced in [10]; they are generalisations of the spaces [w.sup.p.sub.[infinity]] and [w.sup.p.sub.0]. Their dual spaces were determined in [11]. 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.sup.p.sub.[infinity]]([LAMBDA]) or X = [w.sup.p.sub.0]([LAMBDA]) and Y = [w.sup.1.sub.[infinity]] ([LAMBDA]'), for X = [w.sup.1.sub.[infinity]]([LAMBDA]) and Y = [w.sup.p.sub.[infinity]] ([LAMBDA]'), and when X is the [beta]-dual of [w.sup.p.sub.[infinity]]([LAMBDA]) or [w.sup.p.sub.0] ([LAMBDA]) and Y is the [beta]-dual of [w.sup.1.sub.[infinity]] ([LAMBDA]'). As a special case, we obtain the characterisations of the classes of all infinite matrices that map [w.sup.p.sub.[infinity]] into [w.sup.1.sub.[infinity]], and [w.sup.p] into [w.sup.1], the last result being similar to Maddox's and the Silverman--Toeplitz theorems. Furthermore, we prove that the classes ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) 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 w denote the set of all sequences x = [([x.sub.k]).sup.[infinity].sub.k=0], and [l.sub.[infinity]], [c.sub.0] and [empty set] be the sets of all bounded, null and finite complex sequences, respectively; also let cs, bs and

[l.sub.p] = {x [member of] w : [[inifinity].summation over (k=0)] [[[absolute value of [x.sub.k]].sup.p] < [infinity]} for 1 [less than or equal to] p [infinity]

be the sets of all convergent, bounded and absolutely p-summable series. We write e and [e.sup.(n)] (n = 0,1,...) for the sequences with [e.sub.k] = 0 for all k, and [e.sup.(n).sub.n] = 1 and [e.sup.(n).sub.k] = 0 for k [not equal to] n. A sequence ([b.sub.n]) in a linear metric space X is called a Schauder basis of X if for every x [member of] X there exists a unique sequence [[delta].sub.n] of scalars such that x = [summation].n [[delta].sub.n][b.sub.n].

A BK space X is a Banach sequence space with continuous coordinates [P.sub.n](x) = [x.sub.n] (n [member of] [N.sub.0]) for all x [member of] X; a BK space X [contains] [empty set] is said to have AK if x = [infinity].[summation] over k=0 [x.sub.k][e.sup.(k)] for every sequence x = [([x.sub.k]).sup.[infinity].sub.k=0] [member of] X. Let X be a subset of w. Then the set

[X.sup.[beta]] = {a [member of ] w : ax = [([a.sub.k][x.sub.k]).sup.[infinity].sub.k=0] [member of] cs for all x [member of] X}

is called the [beta]-dual of X. Let A = [([a.sub.nk]).sup.[infinity].sub.k=0] be an infinite matrix of complex numbers and x = [([x.sub.k]).sup.[infinity].sub.k=0] [member of] w. Then we write [A.sub.n] = [([a.sub.nk]).sup.[infinity].sub.k=0] (n = 0,1,...) and [A.sup.k] = [([a.sub.nk]).sup.[infinity].sub.k=0] (k = 0,1,...) for the sequences in the n-th row and the k-th column of A, and [A.sub.n]x = [[infinity].summation over (k=0)] provided the series converges. Given any subsets X and Y of w, then (X, Y) denotes the class of all infinite matrices A that map X into Y, that is, [A.sub.n] [member of] [X.sup.[beta]] for all n, and Ax = [([A.sub.n]x).sub.[infinity].sub.n=0] [member of] Y.

Let X and Y be Banach spaces and [B.sub.X] = {x [member of] X : [parallel]x[parallel] [less than or equal to] 1} denote the unit ball in X. Then we write B(X, Y) for the Banach space of all bounded linear operators L : X [right arrow] Y with the operator norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We write [X.sup.*] = B(X,C) for the continuous dual of X with the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all f [member of] [X.sup.*]. The following results and definitions are well known. Since we will frequently apply them, we state them here for the reader's convenience.

Proposition 2.1. Let X and Y be BK spaces.

(a) Then we have (X, Y) [subset] B(X, Y); this means that if A [member of] (X, Y), then [L.sub.A] [member of] B(X, Y), where [L.sub.A](x) = Ax (x [member of] X) (see [14, Theorem 4.2.8]).

(b) If X has AK then we have B(X, Y) [subset] (X, Y); this means every L [member of] B(X, Y) is given by a matrix A [member of] (X, Y) such that L(x) = Ax (x [member] X) (see [4, Theorem 1.9]).

A nondecreasing sequence [LAMBDA] = [([[delta].sub.n]).sup.[infinity].n=0] of positive reals is called exponentially bounded if there is an integer m [greater than or equal to] 2 such that for all nonnegative integers v there is at least one term [[delta].sub.n] in the interval [I.sup.(v).sub.m] = [[m.sup.v], [m.sup.v+1] - 1] ( [10]). It was shown ([10, Lemma 1]) that a nondecreasing sequence [LAMBDA] = [([[delta].sub.n]).sup.[infinity].sub.n=0] is exponentially bounded, if and only if the following condition holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

such a subsequence is called an associated subsequence.

Example 2.2. A simple, but important exponentially bounded sequence is the sequence [LAMBDA] with [[delta].sub.n] = n + 1 for n = 0,1,...; an associated subsequence is given by [[delta].sub.n(v)] = [2.sup.v], v = 0,1,....

Throughout, let 1 [less than or equal to] p < [infinity] and q be the conjugate number of p, that is, q = [infinity] for p = 1 and q = p/(p - 1) for 1 < p < [infinity]. Also let [([[mu].sub.n]).sup.[infinity].sub.n=0] be a nondecreasing sequence of positive reals tending to infinity. Furthermore let [LAMBDA] = [([[lambda].sub.n]).sup.[infinity].sub.n=0] be an exponentially bounded sequence, and [([[lambda].sub.n(v)]).sup.[infinity].sub.v=0] an associated subsequence with [[lambda].sub.n(0)] = [[lambda].sub.0]. We write [K.sup.<v>] (v = 0,1,...) for the set of all integers k with n(v) [less than or equal to] k [less than or equal to] n(v + 1) - 1, and define the sets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If p = 1, we omit the index p throughout, that is, we write [[??].sub.0]([mu]) = [[??].sup.1.sub.[infinity]](mu) etc., for short.

Proposition 2.3 (See [10, Theorem 1 (a), (b)]). Let [([[mu].sub.n]).sup.[infinity].sub.n=0] be a nondecreasing sequence of positive reals tending to infinity, [LAMBDA] = [([[lambda].sub.n]).sup.[infinity].sub.n=0] be an exponentially bounded sequence and [([[lambda].sub.n(v)]).sup.[infinity].sub.n=0] be an associated subsequence.

(a) Then [[??].sup.p.sub.0]([mu]) and [[??].sup.p.sub.[infinity]]([mu]) are BK spaces with the sectional norm [[parallel]*[parallel].sup.~.sub.[mu]] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[??].sup.p.sub.0]([mu]) has AK.

(b) We have [[??].sup.p.sub.0]([LAMBDA]) = [w.sub.p.sub.0]([LAMBDA]), [[??].sub.p.sub.[infinity]]([LAMBDA]) = [w.sub.p.sub.[infinity]]([LAMBDA]), and the sectional norm [[parallel]*[parallel].sub.[LAMBDA]] and the block norm [[parallel]*[parallel].sup.~.sub.[LAMBDA]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are equivalent on [w.sup.p.sub.0] ([LAMBDA]) and on [w.sup.p.sub.[infinity]]([LAMBDA]).

Remark 2.4. (a) It can be shown that [w.sup.p.sub.[infinity]]([LAMBDA]) is not separable, and so has no Schauder basis.

(b) It follows from [14, Corollary 4.2.4] and Proposition 2.3, that [w.sup.p.sub.0]([LAMBDA]) and [w.sup.p.sub.[infinity]]([LAMBDA]) are BK spaces with the norm [[parallel]*[parallel].sub.[LAMBDA]] and that [w.sup.p.sub.0]([LAMBDA]) has AK.

Example 2.5. We might also define the set

[w.sup.p]([LAMBDA]) = [w.sup.p.sub.0]([LAMBDA]) [direct sum] e = {x [member of] w : x - [xi] [member of] [w.sup.p.sub.0] for some complex number [xi]}.

It can be shown that the strong [LAMBDA]-limit [xi] of any x [member of] [w.sup.p]([LAMBDA]) is unique if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and that [w.sup.p]([LAMBDA]) [subset] [w.sup.p.sub.[infinity]]([LAMBDA]) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of Proposition 2.3 (b) and Example 2.2, the sets [w.sup.p.sub.0]([LAMBDA]) and [w.sup.p.sub.[infinity]] ([LAMBDA]) reduce to the BK spaces [w.sup.p.sub.0] and [w.sup.p.sub.[infinity]] for [[delta].sub.n] = n +1 for n = 0,1,...; it is also clear that then [bar.[LAMBDA]] < [infinity] and [bar.[lambda]] > 0, and consequently [w.sup.p] is a BK space and the strong limit [xi] of each sequence x [member of] [w.sup.p] is unique.

Throughout, we write [parallel]*[parallel] = [[parallel]*[parallel].sub.[LAMBDA]], for short.

The [beta]-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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] provided the expression on the right exists and is finite, which is the case whenever X is a BK space and a [member of] [X.sup.[beta]] by [14, Theorem 7.2.9]. If A is an exponentially bounded sequence with an associated subsequence [[lambda].sub.n(v)], then we write max and [summation over (v)] for the maximum and sum taken over all k [member of] [K.sup.<v>]. We denote by [x.sup.<v>] = [v.summation over (v)] [x.sup.k][e.sup.(k)] (v [member of] [N.sub.0]) the v -block of the sequence x.

Parts (a) and (b) of the next result are [11, Theorem 5.5 (a), (b)], Part (c) is [11, Theorem 5.7], and Parts (d) and (e) are [11, Theorem 5.8 (a), (b)].

Proposition 2.6. Suppose [LAMBDA] = [([[lambda].sub.n]).sup.[infinity].sub.n=0] is an exponentially bounded sequence and let [([[lambda].sub.n(v])).sup.[infinity].sub.v=0] be an associated subsequence. We write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(a) Then we have [([w.sup.p]([LAMBDA])).sup.[beta]] = [([w.sup.p.sub.[infinity]]([LAMBDA]).sup.[beta]] = [M.sub.p]([LAMBDA]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

(b) The continuous dual [w.sup.p.sub.0]([LAMBDA])* of [w.sup.p.sub.0]([LAMBDA]) is norm isomorphic to [M.sub.p]([LAMBDA]) with the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(c) Then [M.sub.p]([LAMBDA]) is a BK space with AK with respect to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] .

(d) We have [([w.sup.p.sub.[infinity]]([LAMBDA])).sup.[beta][beta]] = [([w.sup.p.sub.0][LAMBDA])).sup.[beta][beta]] = [w.sup.p.sub.[infinity]]([LAMBDA]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

(e) The continuous dual ([M.sub.p]([LAMBDA]))* of [M.sub.p]([LAMBDA]) is norm isomorphic to [w.sup.p.sub.[infinity]]([LAMBDA]).

Remark 2.7. ([LAMBDA]) The continuous dual of [w.sub.[infinity]]([LAMBDA]) is not given by a sequence space.

(b) The set [w.sup.p.sub.[infinity]]([LAMBDA]) is [beta]-perfect, that is, ([w.sup.p.sub.[infinity]]([LAMBDA]))[beta][beta] = [w.sup.p.sub.[infinity]]([LAMBDA]).

3 Matrix Transformations on [w.sup.p.sub.[infinity]]([LAMBDA]) and [w.sup.p.sub.0]([LAMBDA])

Let [LAMBDA] = [([[lambda].sub.k]).sup.[infinity].sub.k=0] and [LAMBDA]' = [([[lambda]'.sub.m]).sup.[infinity].sub.m=0] be exponentially bounded sequences and [([[lambda].sub.k(v)]).sup.[infinity].sub.v=0] and [([[lambda]'.sub.m(mu)]).sup.[infinity].sub.[mu]=0] be associated subsequences. Furthermore, let [K.sup.<v>] (v = 0,1,...) and [M.sup.<[mu]>] ([mu] = 0,1,...) be the sets of all integers k and m with k(v) [less than or equal] k [less than or equal to] k(v + 1) - 1 and m([mu]) [less than or equal to] m [less than or equal to] m([mu] + 1) - 1. If A = [([a.sub.mk]).sup.[infinity].sub.m,k=0] is an infinite matrix and M = [([M.sub.[mu]]).sup.[infinity].sub.[mu]=0] is a sequence of subsets [M.sub.[mu]] of [M.sup.<[mu]>] for [mu] = 0,1,..., we write [S.sup.M](A) for the matrix with the rows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we establish necessary and sufficient conditions for an infinite matrix A to be in the classes ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')), ([w.sup.p.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')) and ([M.sub.p]([LAMBDA]), M([LAMBDA]')), and consider the special cases of ([w.sup.p.sub.[infinity]],[w.sub.[infinity]]) and ([w.sup.p], w). We also estimate the operator norms of [L.sub.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.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')) and ([w.sup.p.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')), and estimate the operator norm of [L.sub.A] when the matrix A is a member of those classes.

Theorem 3.1. Let [LAMBDA] = [([[lambda].sub.k]).sup.[infinity].sub.k=0] and [LAMBDA]' = [([lambda]'m).sup.[infinity].sub.m=0] be exponentially bounded sequences and [([[lambda].sub.k(v)]).sup.[[infinity].sub.v=0] and [([[lambda]'.sub.m(mu)]).sup.[infinity].sub.v=0] be associated subsequences. Then we have A [member of] ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (3.1)

moreover, we have ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')) = ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')). If A [member of] ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')), then the operator norm of [L.sub.A] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Proof. Throughout the proof, we write [parallel]A[parallel] = [[parallel]A[parallel].sub.([LAMBDA],[LAMBDA]')], for short.

First we assume that the condition in (3.1) is satisfied. Let m [member of] [N.sub.0] be given. Then there is a unique [[mu].sub.m] [member of] [N.sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and it follows from (3.1) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], that is, [A.sub.m] [member of] [([w.sup.p.sub.[infinity]]([LAMBDA])).sup.[beta]] by Proposition 2.6 (a). Thus we have shown [A.sub.m] [member of] [([w.sub.[infinity]]([LAMBDA])).sup.[beta]] for all m [member of] [N.sub.0]. Now let x [member of] [w.sup.p.sub.[infinity]]([LAMBDA]) be given. For each [mu] [member of] [N.sub.0], we write [M.sub.[mu](x)] for a subset of [M.sup.<[mu]>] for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and put M(x) = [([M.sub.[mu](x)]).sup.[infinity].sub.[mu]=0]. Then we have by a well-known inequality (see [12]), (2.1) and (3.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.3)

and consequently Ax [member of] [w.sub.[infinity]]([LAMBDA]') for all x [member of] [w.sup.p.sub.[infinity]]([LAMBDA]). Thus, we have shown that if the condition in (3.1) is satisfied, then A [member of] ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')) [subset] ([w.sup.p.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')).

Conversely, we assume A [member of] ([w.sup.p.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')). Then we have [A.sub.m] [member of] [([w.sup.p.sub.0]([LAMBDA])).sup.[beta]] for all m [member of] [N.sub.0], hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m by Proposition 2.6 ([LAMBDA]). Since [w.sup.p.sub.0]([LAMBDA]) and [w.sub.[infinity]]([LAMBDA]') are a BK spaces by Remark 2.4 (b), it follows from Proposition 2.1 (a) that [L.sub.A] [member of] B([w.sup.p.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')), and so [parallel][L.sub.A][parallel] < [infinity]. We also have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [M.sub.[mu]] [subset] [M.sup.<[mu]>] and all [mu] [member of] [N.sub.0], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x [member of] [w.sup.p.sub.0]([LAMBDA]). Since trivially [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x [member of] [w.sup.p.sub.0]([LAMBDA]), all [M.sub.[mu]] [subset] [M.sup.<[mu]>] and all [mu] [member of] [N.sub.0], it follows by (2.1) in Proposition 2.6 (a) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [M.sub.[mu]] [subset] [M.sup.<[mu]>] and [mu] [member of] [N.sub.0], and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Thus we have shown that if A [member of] ([w.sup.p.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')), then (3.3) is satisfied.

It remains to show that if A [member of] ([w.sup.p.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]')), 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.sub.p]([LAMBDA]), [MU]([LAMBDA]')), and estimate the operator norm of [L.sub.A] when A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')). We write T for the set of all sequences t = [([t.sub.[mu]]).sup.[infinity].sub.[mu]=0] such that for each [mu] = 0,1,... there is one and only one [t.sub.[mu]] [member of] [M.sup.<[mu]>].

Theorem 3.2. Let [LAMBDA] = [([[delta].sub.k]).sup.[infinity].sub.k=0] and [LAMBDA]' = [([[delta]'.sub.m]).sup.[infinity].sub.m=0] be exponentially bounded sequences and [([[delta].sub.k(v)]).sup.[infinity].sub.v=0] and [([[delta]'.sub.m(mu)]).sup.[infinity].sub.[mu]=0] be associated subsequences. Then we have A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

where, of course,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')), then the operator norm of [L.sub.A] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Proof. Throughout the proof, we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for short.

First we assume that the condition in (3.5) is satisfied. Let m [member of] [N.sub.0] be given. Then there is a unique [[mu].sub.m] [member of] [N.sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We choose N = {m} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then it follows from (3.5) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and so [A.sub.m] [member of] [w.sup.p.sub.[infinity]]([LAMBDA]) = [([M.sub.p]([LAMBDA])).sup.[beta]] by Proposition 2.6 (a) and (d). Now let [[mu].sub.0] [member of] [N.sub.0] and x [member of] [M.sub.p]([LAMBDA]) be given. For each [mu] [member of] [N.sub.0] with 0 [less than or equal to] [mu] [less than or equal to] [[mu].sub.0], let m([mu]; x) be the smallest integer in [M.sup.<[mu]>] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [absolute value of [A.sub.m]x] = [absolute value of [A.sub.m([mu];x)]x]. Then we have by a well-known inequality (see [12]) and (2.2) in Proposition 2.6 (d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [[mu].sub.0] [member of] [N.sub.0] was arbitrary, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

and consequently Ax [member of] M([LAMBDA]') for all x [member of] [M.sub.p]([LAMBDA]). Thus we have shown that if the condition in (3.5) is satisfied, then A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')).

Conversely, we assume A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')). Then [A.sub.m] [member of] ([[M.sub.p]([LAMBDA])).sup.[beta]] = [w.sup.p.sub.[infinity]]([LAMBDA]) for all m [member of] [N.sub.0] by Proposition 2.6 (a) and (d). Furthermore, since [M.sub.p]([LAMBDA]) and M([LAMBDA]') are BK spaces by Proposition 2.6 (c), it follows from Proposition 2.1 (a) that [L.sub.A] [member of] [BETA]([M.sub.p]([LAMBDA]), M([LAMBDA]')). We also have [L.sub.N,t] [member of] ([M.sub.p]([LAMBDA]))* for all finite subsets N of [N.sub.0] and all sequences t [member of] T, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x [member of] [M.sub.p]([LAMBDA]). Since trivially [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all finite subsets N of [N.sub.0] and all t [member of] T, it follows by (2.2) in Proposition 2.6 (c) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since this holds for all finite subsets N of [N.sub.0] and all t [member of] T, we conclude

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

Thus we have shown that if A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')), then (3.5) is satisfied.

Finally, if A [member of] ([M.sub.p]([LAMBDA]), M([LAMBDA]')), then (3.6) follows from (3.8) and (3.7). []

Using the transpose [A.sup.T] of a matrix A, we obtain an alternative characterisation of the class ([w.sub.[infinity]]([LAMBDA]),[w.sup.p.sub.[infinity]]([LAMBDA]')).

Theorem 3.3. We have A [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sup.p.sub.[infinity]] ([LAMBDA]')) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

Proof. Since X = [w.sub.0]([LAMBDA]) and Z = [M.sub.p]([LAMBDA]') are BK spaces with AK by Remark 2.4 (b) and Proposition 2.6 (c), and Y = [Z.sup.[beta]] = [w.sup.p.sub.[infinity]]([LAMBDA]') by Proposition 2.6 (d), it follows from [14, Theorem 8.3.9] that A [member of] ([w.sub.0]([LAMBDA]), [w.sup.p.sub.[infinity]]([LAMBDA]')) = (X, Y) = ([X.sup.[beta][beta]], Y) = ([w.sub.[infinity]]([LAMBDA]), [w.sup.p.sub.[infinity]]([LAMBDA]')) and A [member of] (([w.sub.0]([LAMBDA]), [w.sup.p.sub.[infinity]]([LAMBDA]')) if and only if [A.sup.T] [member of] (Z,[X.sup.[beta]]) = ([M.sub.p]([LAMBDA]'), M([LAMBDA])), 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.sup.p.sub.[infinity]], [w.sub.[infinity]]), ([w.sup.p], w) and ([w.sub.[infinity]], [w.sup.p.sub.[infinity]]). Let [LAMBDA] = [LAMBDA]' and [[lambda].sub.n] = n +1 for n = 0,1,... as in Examples 2.2 and 2.5. Then we may choose the subsequences given by [[lambda].sub.k(v)] = [2.sup.v] and [[lambda].sub.m([mu])] = [2.sup.[mu]] for all v, a = 0,1,..., and consequently the sets [K.sup.<v>] and [M.sup.<[mu]>] are the sets of all integers k and m with [2.sup.v] [less than equal to] k [less than or equal to] [2.sup.v+1] - 1 and [2.sup.[mu]] [less than or equal to] m [less than or equal to] [2.sup.[mu]+1] - 1. We also write [M.sub.p] = [M.sub.p]([LAMBDA]).

Remark 3.4. (a) We obviously have [w.sup.p.sub.0] [subset] [w.sup.p] [subset] [w.sup.p.sub.[infinity]] .

(b) For each x [member of] [w.sup.p], the strong limit [xi], that is, the complex number [xi] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is unique (see [5]).

(c) Every sequence x = [([x.sub.k]).sup.[[infinity].sub.k=0] [member of] [w.sup.p] has a unique representation

x = [xi] x e + [[infinity].summation over (k=0)] ([x.sub.k] - [xi]) [e.sup.(k)] ([5]).

Example 3.5. (a) It follows from Theorem 3.1 that A [member of] ([w.sup.p.sub.[infinity]], [w.sub.[infinity]]) = ([w.sup.p.sub.0], [w.sub.[infinity]]) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) It follows from Part (a) and [14, 8.3.6, 8.3.7] that A [member of] ([w.sup.p],w) if and only if (3.10),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

hold.

(c) We obtain from Theorems 3.2 and 3.3, interchanging the roles of N and K, and [mu] and v, that A [member of] ([w.sub.[infinity]], [w.sup.p.sub.[infinity]]) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We also give a formula for the strong limit of Ax when A [member of] ([w.sup.p], w) and x [member of] [w.sup.p].

Theorem 3.6. If A [member of] ([w.sup.p], w), then the strong limit [eta] of Ax for each sequence x [member of] [w.sup.p] is given by

[eta] = [??] x [xi] + [[infinity].summation over (k=0)] [[alpha].sub.k]([x.sub.k] - [xi]) (3.13)

where [xi] is the strong limit of the sequence x, and the complex numbers [??] and [[alpha].sub.k] for k = 0,1,... are given by (3.12) and (3.11) in Example 3.5 (b).

Proof. We assume A [member of] ([w.sup.p], w) and write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for short. The complex numbers [??] and [[alpha].sub.k] for k = 0,1,... exist by Example 3.5 (b).

First, we show [([[alpha].sub.k]).sup.[infinity].sub.k=0] [member of] [M.sub.p]. Let x [member of] [w.sup.p] and [k.sub.0] [member of] [N.sub.0] be given. Then there exists an integer v([k.sub.0]) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and we have by the inequality in [9, Lemma 1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Letting [mu] tend to [infinity0], we obtain [[k.sub.0].summation over (k=0)] [absolute value of [[alpha].sub.k][x.sub.k]x] [less than or equal to] 0 + 4 x [parallel]A[parallel] < [infinity] from (3.11) and (3.10).

Since [k.sub.0] [member of] [N.sub.0] was arbitrary, it follows that [[infinity].summation over (k=0)] [absolute value of [[alpha].sub.k][x.sub.k]] < [infinity] for all x [member of] [w.sup.p], that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we write [??](x) = [[infinity].summation over (k=0)] [[alpha].sub.k][x.sub.k] and B = [([b.sub.nk]).sup. [[infinity].sub.k=0] for the matrix with [b.sub.nk] = [[alpha].sub.nk] - [[alpha].sub.k] for all n and k, and show

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.14)

Let x [member of] [w.sup.p.sub.0] and [epsilon] > 0 be given. Since [w.sup.p.sub.0] has AK, there is [k.sub.0] [member of] [N.sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It also follows from (3.11) that there is [[mu].sub.0] [member of] [N.sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [mu] [greater than or equal to] [[mu].sub.0] be given. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus we have shown (3.14).

Finally, let x [member of] [w.sup.p] be given. Then there is a unique complex number [xi] such that [x.sup.(0)] = x - [xi] x e [member of] [w.sup.p.sub.0], by Remark 3.4 (b), and we obtain by (3.14) and (3.12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof. []

4 The Banach Algebra ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]))

In this section, we show that ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) is a Banach algebra with respect to the norm [parallel]*[parallel] defined by [parallel]A[parallel] = [parallel][L.sub.A][parallel] for all A [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])). We also consider the nontrivial special case of (w, w). We need the following results.

Lemma 4.1. (a) The matrix product B x A is defined for all A, B G (w[infinity]([LAMBDA]), w[infinity]([LAMBDA])); in fact

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.1)

(b) Matrix multiplication is associative in ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])).

(c) The space ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) is a Banach space with respect to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

Proof. (a) Let A, B [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])). First we observe that [e.sup.(k)] [member of] [w.sub.[infinity]]([LAMBDA]) implies [Ae.sup.(k)] = [([A.sub.m][e.sup.(k)]).sup.[infinity].sub.m=0] = [([a.sub.mk]).sup.[infinity].sub.m=0] = [A.sup.k] [member of] [w.sub.[infinity]]([LAMBDA]) for all k. Therefore we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

Furthermore B [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) implies [B.sub.n] [member of] [([w.sub.[infinity]]([LAMBDA])).sup.[beta]] = M([LAMBDA]) for all n, that is, by Proposition 2.6 (a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

Now it follows from (4.3) and (4.4) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) Let A,B,C [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])). We write for D [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA]))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and note that [M.sup.T](D) < [infinity] by Theorem 3.3. We are going to show that the series [[infinity].summation over (k=0)] [[infinity].summation over (m=0)] [a.sub.nm][b.sub.mk][c.sub.kj] are absolutely convergent for all n and j. We fix n and j and write s = [A.sub.n] and t = [C.sup.j] for the sequences in the n-th row of A and the j-th column of C. Then we have s [member of] [MU]([LAMBDA]) and t [member of] [w.sub.[infinity]]([LAMBDA]). We define the matrix D = [([d.sub.[mu]k]).sup.[[infinity].sub.k=0] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, given [mu] [member of] [N.sub.0], for every v = 0,1,..., let [k.sub.v] = [k.sub.v]([LAMBDA]) [member of] [K.sup.<v>] be the smallest integer with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then by the inequality in [9, Lemma 1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

It also follows that for [mu] = 0, 1,...

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

Therefore, we obtain from (4.6) and (4.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we have shown that [[infinity].summation over (m=0)] [[infinity].summation over (k=0)] [s.sub.m][b.sub.mk][t.sub.k] is absolutely convergent, and consequently matrix multiplication is associative in ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])).

(c) We assume that [([A.sup.(j)]).sup.[infinity].sub.j=0] is a Cauchy sequence in ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])). Since ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) = ([w.sub.0]([LAMBDA]), [w.sub.0]([LAMBDA])) by Theorem 3.1 and [w.sub.0]([LAMBDA]) has AK by Remark 2.4 (b), it is a Cauchy sequence in ([w.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) = [BETA]([w.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])), by Proposition 2.1. Consequently there is [L.sub.A] [beta] [BETA]([w.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) with [L.sub.[A.sup.(j)]] [right arrow] [L.sub.A]. Since [w.sub.0]([LAMBDA]) has AK there is a matrix A [member of] ([w.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) by Proposition 2.1 (b) such that Ax = [L.sub.A](x) for all x [member of] [w.sub.0]([LAMBDA]). Finally ([w.sub.0]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) = ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) implies A [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])). []

The following result is obtained as an immediate consequence of Lemma 4.1.

Theorem 4.2. The class ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])) is a Banach algebra with respect to the norm [parallel]A[parallel] = [parallel][L.sub.A][parallel] for all A [member of] ([w.sub.[infinity]]([LAMBDA]), [w.sub.[infinity]]([LAMBDA])).

The following example is obtained from Theorem 4.2.

Example 4.3. Let [[delta].sub.n] = n + 1 for n = 0,1,... as in Examples 2.2, 2.5 and 3.5. Then ([w.sub.[infinity]], [w.sub.[infinity]]) is a Banach algebra with [parallel]A[parallel] = [parallel]LA[parallel].

Finally, we show that (w, w) is a Banach algebra.

Theorem 4.4. The class (w, w) is a Banach algebra with [parallel]A[parallel] = [parallel][L.sub.A][parallel].

Proof. We have to show in view of Theorem 4.2 that

(i) (w, w) is complete;

(ii) if A, B [member of] (w, w), then B x A [member of] (w, w).

First we show (i). Let [([A.sup.(j)]).sup.[infinity].sub.j=1] be a Cauchy sequence in (w,w). Since (w,w) [subset] ([w.sub.[infinity]], [w.sub.[infinity]]) and the operator norm on B([w.sub.[infinity]], [w.sub.[infinity]]) is the same as that on B(w,w), it follows that [([A.sup.(j)]).sup.[infinity].sub.j=1] is a Cauchy sequence in ([w.sub.[infinity]], [w.sub.[infinity]]), and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [A.sup.(j)] by Lemma 4.1 (c). We have to show A [member of] (w, w). Let [epsilon] > 0 be given. Since [([A.sup.(j)]).sup.[infinity].sub.j=1] is a Cauchy sequence in (w, w) there exists a [j.sub.0] [member of] [N.sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.7)

Also, by (3.11) and (3.12), for each fixed j there exist complex numbers [[alpha].sup.(j).sub.k] (k = 0,1,...) and [[??].sup.(j)] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.9)

Let j, l [greater than or equal to] [j.sub.0] be given. Then we have for each fixed k [member of] [N.sub.0] by (4.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Letting [mu] [right arrow] [infinity], we obtain from (4.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus [([[alpha].sup.(j).sub.k]).sup.[infinity].sub.j=1] is a Cauchy sequence of complex numbers for each fixed k [member of] [N.sub.0] and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.10)

Now let k [member of] [N.sub.0] be fixed. Then we obtain for all sufficiently large j and for all [mu] by (4.10) and since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Letting [mu] [right arrow] [infinity], we obtain from (4.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [epsilon] > 0 was arbitrary, it follows that [[alpha].sub.k] satisfies the condition in (3.11) of Example 3.5 (b). Using exactly the same argument as before with [a.sup.(j).sub.nk] and [[alpha].sup.(j).sub.k] replaced by [[infinity].summation over (k=0)] [a.sup.(j).sub.nk] and [[??].sup.(j)], and applying (4.9) instead of (4.8), we conclude that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists and satisfies the condition in (3.12) of Example 3.5 (b). Finally A [member of] ([w.sub.[infinity]], [w.sub.[infinity]]) and (3.11) and (3.12) imply A [member of] (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 [member of] (w, w) implies B x A [member of] (w, w). Since A, B [member of] (w, w), by Example 3.5 (b), there are complex numbers [[alpha].sub.k], [??] that satisfy (3.11) and (3.12), and complex numbers [[beta].sub.k], [beta] that satisfy (3.11) and (3.12) with [b.sub.nk], [[??].sub.k] and [??] instead of [a.sub.nk], [??]k and [??]. Let x [member of] w be given and [xi] be the strong limit of x. We put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We observe that [([[alpha].sub.k]).sup.[[infinity].sub.k=0], [([[beta].sub.n]).sup.[infinity].sub.n=0] [member of] [MU] by the proof of Theorem 3.6, and also trivially M [subset] [l.sub.1] [subset] cs. Therefore all the series in the definition of [zeta] converge. We write C = B x A, y = Ax, [eta] for the strong limit of the sequence y, and [zeta]' for the strong limit of the sequence z = By. Since Cx = B (Ax) by Lemma 4.1 (b), we obtain by (3.13) in Theorem 3.6

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This shows that Cx [member of] w, and completes the proof of (ii). []

References

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[3] I. Djolovic, E. Malkowsky, A note on Fredholm operators on ([c.sub.0])T Applied Mathematics Letters, 22:1734-1739, 2009.

[4] A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat, 17:59-78, 2003.

[5] I. J. Maddox, On Kuttner's theorem, J. London Math. Soc., 2(1):316-322, 1968.

[6] B. de Malafosse, E. Malkowsky, Sequence spaces and inverse of an infinite matrix, Rend. del Circ. Mat. di Palermo, Serie II 51:277-294, 2002.

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[8] B. de Malafosse, V. Rakocevic, Application of measure of noncompactness in operators on the spaces [s.sub.[alpha]], [s.sup.0.sub.[alpha]], [s.sup.(c).sub.[alpha]], [l.sup.p.sub.[alpha]], J. Math. Anal. Appl., 323(1):131-145, 2006.

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[12] A. Peyerimhoff, Uber ein Lemma von Herrn Chow, J. London Math. Soc., 32:33-36, 1957.

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Eberhard Malkowsky

Fatih University

Department of Mathematics

34500 Buyukcekmece, Istanbul, Turkey

emalkowsky@fatih.edu.tr

eberhard.malkowsky@math.uni-giessen.de

Received September 30, 2010; Accepted November 22, 2010

Communicated by Malisa R. Zizovic
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