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

1 Introduction

Maddox  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 .

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 ; they are generalisations of the spaces [w.sup.p.sub.[infinity]] and [w.sup.p.sub.0]. 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.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] ( ). 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 ), (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 ) 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 ).

(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)] ().

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). []

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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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Author: Printer friendly Cite/link Email Feedback Malkowsky, Eberhard Advances in Dynamical Systems and Applications Report 7TURK Jun 1, 2011 7102 Matrix transformations and statistical convergence II. Preface. Banach spaces Matrices Matrices (Mathematics) Sequences (Mathematics) Transformations (Mathematics)