# A note on families of generalized Norlund matrices as bounded operators on [l.sub.p]/Monest uldistatud Norlundi maatriksite perest, kus maatriksid on tokestatud operaatorid ruumis [l.sub.p].

1. INTRODUCTION AND PRELIMINARIES

1.1. Suppose throughout the paper that

1 < p < [infinity], 1/p + 1/q = 1

Suppose also that A = ([a.sub.nk]) is a triangular matrix of non-negative real numbers, that is, [a.sub.nk] [greater than or equal to] 0 for n,k [greater than or equal to] 0, and [a.sub.nk] = 0 for n > k, n,k [member of] [N.sub.0]. Let [l.sub.p] be the Banach space of all complex sequences x = ([x.sub.n]) (n [member of] [N.sub.0]) with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and let B([l.sub.p]) be the Banach algebra of all bounded linear operators on [l.sub.p]. Thus A [member of] B([l.sub.p]) if and only if Ax [member of] [l.sub.p] whenever x [member of] [l.sub.p], where Ax = ([y.sub.n]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that A [member of] B([l.sub.p]) if and only if [parallel]A[[parallel].sub.p] < [infinity], in which case [parallel]A[[parallel].sub.p] is the norm of A.

It is well known that A is a bounded operator on the Banach space m of bounded sequences if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This condition, together with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is necessary and sufficient for A to be a bounded operator on the Banach space [c.sub.0]. But even on these two conditions A need not be a bounded operator on [l.sub.p]. As an example the Norlund method [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with 0 < [phi] < 1 can be given (see [5]). Also, the Riesz weighted mean matrix A = ([bar.N], 1/n+1]) is not a bounded operator on [l.sub.p] because the necessary condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for A to be bounded on lp is not satisfied for it.

1.2. The problem of characterizing matrices in B([l.sub.p]) by means of conditions that are not complicated and difficult to apply has been discussed in a number of papers. This problem was discussed, for example, by D. Borwein and other mathematicians in papers [3,7,8] in general and, in particular, for Norlund, Riesz weighted mean and Hausdorff matrices in [1-6,10,12]. In these papers different types of conditions (mostly sufficient) for A to be in B([l.sub.p]) were proved and illustrated with examples, also estimates for the norm [parallel]A[[parallel].sub.p] were found. It should be mentioned that already in 1943 G. H. Hardy proved (see [11]) an inequality which says that the Cesaro matrices A = (C, [alpha]) and the Euler-Knopp matrices A = (E, [alpha]) ([alpha]> 0) are in B([l.sub.p]) and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] respectively.

1.3. We consider in our paper generalized Norlund matrices.

Suppose throughout the paper that ([p.sub.n]) and ([q.sub.n]) are two non-negative sequences such that [p.sub.0], [q.sub.0] > 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let us consider the qeneralized Norlund matrix A = (N, [P.sub.n, [q.sub.n]), i.e., the matrix A = ([a.sub.nk]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In particular, if [q.sub.n] = 1 for any n [member of] [N.sub.0], then we get the Norlund matrix (N, [p.sub.n], 1) = (N, [p.sub.n]). If [p.sub.n] = 1 for any n [member of] [N.sub.0], then we get the Riesz matrix ([bar.N], 1, [q.sub.n]) = (N, [q.sub.n]). In particular, if [p.sub.n] = [[alpha].sp.n]/[n.sup.!] and [q.sub.n] = 1/[n.sup.!], we have the Euler-Knopp matrices (N, [p.sub.n], [q.sub.n]) = (E, [alpha]).

The most convenient conditions to show that the matrix A = (N, [p.sub.n], [q.sub.n]) is in B([l.sub.p]) come from the following theorem of D. Borwein (see [3], Theorem 2) proved for A = ([a.sub.nk]).

Theorem A. Suppose that A = ([a.sub.nk]) satisfies the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

and

[a.sub.nk] [less than or equal to] [M.sub.2][a.sub.nj] for 0 [less than or equal to] k [less than or equal to] j [less than or equal to] n, (1.2)

where [M.sub.2] is a positive number independent of k, j,n.

Then A [member of] B([l.sub.p]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.3)

where [lambda]= lim inf [na.sub.n0].

Notice that (N, [p.sub.n], [q.sub.n]) satisfies (1.1) with [M.sub.1] = 1. Thus Theorem A gives the following immediate corollary.

Corollary 1. If ([p.sub.n]) is non-increasing and ([q.sub.n]) is non-decreasing, then A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) and (1.3) holds with [a.sub.00] = [M.sub.1] = [M.sub.2] = 1.

Example 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] by Corollary 1.

1.4. The main idea of our paper is to show that on the basis of a given matrix A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) the families of matrices [[A.sub.[alpha]] being in B([l.sub.p]) can be constructed, where [alpha] is a continuous or discrete parameter. Proving Theorems 1 and 2, we will find out some families of matrices [A.sub.[alpha]] = (N, [A.sup.[alpha].sub.n], [q.sub.n]) (see, e.g., [19] and [13]) which are in B([l.sub.p]) if (N, [p.sub.n], [q.sub.n]) is in B([l.sub.p]). It should be mentioned that if A = (N, [p.sub.n], [q.sub.n]) satisfies the conditions of Corollary 1, then the matrices [A.sub.[alpha]] [member of] B([l.sub.p]) in Theorems 1 and 2 need not satisfy these conditions any more. In other words, ([p.sup.[alpha].sub.n]) need not be non-increasing any more (if ([p.sub.n]) is), but nevertheless [A.sub.[alpha]] are bounded operators on [l.sub.p].

1.5. We need also the preliminaries below.

The following theorem was published by Borwein in [3] as Theorem 1.

Theorem B. Suppose that A = ([a.sub.nk]) satisfies conditions (1.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.4)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.5)

Then A [member of] B([l.sub.p]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.7)

and

[[micro].sub.2] [less than or equal to] [M.sub.4] + q[M.sub.3]. (1.8)

We will use also the following simple proposition.

Proposition A. Let [A.sub.1] and [A.sub.2] be two matrices and A = [A.sub.2] [A.sub.1] their product. If [A.sub.1] [member of] B([l.sub.p]) and [A.sub.2] [member of] B([l.sub.p]), then also A [member of] B([l.sub.p]) and

[parallel]A[[parallel].sub.p] [less than or equal to] [parallel]A[[parallel].sub.2][parallel]A[[[parallel].sub.1].sub. p]

2. SOME REMARKS ON GENERALIZED NORLUND MATRICES (N, [p.sub.n], qn) AS BOUNDED OPERATORS ON [l.sub.p]

2.1. First we notice that Corollary 1 can be slightly generalized. If ([p.sub.n]) satisfies the condition

[C.sub.1] [a.sub.n] [less than or equal to] [p.sub.n] [less than or equal to][C.sub.2][a.sub.n] (n [member of] [N.sub.0]), (2.1)

where ([b.sub.n]) is some non-negative sequence and [C.sub.1] and [C.sub.2] are positive numbers not depending on n, we write [p.sub.n] [aproximately equal to] [a.sub.n]. If, in addition, ([a.sub.n]) is non-decreasing, then ([p.sub.n]) is said to be almost non-decreasing. If [p.sub.n] [aproximately equal to] [a.sub.n] and ([a.sub.n]) is non-increasing, then (pn) is said to be almost non-increasing.

Thus, if

[D.sub.1] [b.sub.n] [less than or equal to] [q.sub.n] [less than or equal to] [D.sub.2] [b.sub.n] (n [member of] [N.sub.0]), (2.2)

where ([b.sub.n]) is some non-decreasing sequence and [D.sub.1] and [D.sub.2] are positive constants, then ([q.sub.n]) is almost non-decreasing.

Now the following corollary from Theorem A improves Corollary 1.

Corollary 2. Suppose that ([p.sub.n]) is almost non-increasing and ([q.sub.n]) is almost non-decreasing, i.e., that (2.1) and (2.2) hold with some non-increasing ([a.sub.n]) and non-decreasing ([b.sub.n]), respectively. Then A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) and the estimate in (1.3) for the norm [parallel]A[[parallel].sub.p] is valid with [M.sub.1] = [a.sub.00] = 1 and [M.sub.2] = [C.sub.2][D.sub.2]/[C.sub.1][D.sub.1].

Proof. We have the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for any n [less than or equal to] j. Thus condition (1.2) is satisfied and our statement is true by Theorem A.

Example 2. If [p.sub.n] = [[[alpha].sup.n]/n!] and [q.sub.n] = log(n+2), where [alpha] > 0, then (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) by Corollary 2 because ([p.sub.n]) is almost non-increasing.

2.2. Applying Theorem B to (N, [p.sub.n], [q.sub.n]), we get the following result. Corollary 3. Suppose that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII. (2.3)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Then A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) and the norm [parallel]A[[parallel].sub.p] satisfies (1.6), where

[[micro].sub.1] [less than or equal to] [2.sup.1/p] + 2q[K.sub.1][K.sub.2] (2.5)

and

[[micro].sub.2] [less than or equal to] [K.sub.1]+2q[K.sub.1][K.sub.2]. (2.6)

Proof. Let us show that conditions (1.1), (1.4), and (1.5) are satisfied. We know that (1.1) is satisfied with [M.sub.1] = 1. Further, with the help of (2.3) we get,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus, (1.5) is satisfied with [M.sub.4] [less than or equal to] [K.sub.1], Finally, using (2.3) and (2.4), we get for all 0 [less than or equal to] k [less than or equal to] n/2 .

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus also (1.4) is satisfied with [M.sub.3] [less than or equal to] 2[K.sub.1][K.sub.2]. So we have by Theorem B that inequality (1.6) holds together with (2.5) and (2.6), which come from (1.7) and (1.8), respectively.

We add some remarks to Corollary 3.

Remark 1. In particular, if [q.sub.n] = 1 for all n [member of] [N.sub.0], then (2.3) is satisfied and [K.sub.1] = 1. For this partial case Corollary 3 was proved in [3] as Example 1.

Example 3. If [p.sub.n] = 1 (n [member of] [N.sub.0]) and [q.sub.n] = { 1 if n is even, 0 if nis odd,

then A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) by Corollary 3 because conditions (2.3) and (2.4) are satisfied.

Example 4. Suppose that [p.sub.n] [aproximately equal to][n.sup.[alpha]-1][L.sub.1](n) and [q.sub.n] [n.sup.[sigma][L.sub.2](n), where [alpha]>0, d[greater than or equal to]0, [L.sub.1](*) and [L.sub.2](*) are slowly varying functions and [L.sub.2](*) is non-decreasing. Let us show that A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]). We have that ([q.sub.n]) is almost non-decreasing,

[r.sub.n] [aproximately equal to] [n.sup.[alpha]]+[sigma]] [L.sub.1](n)[L.sub.2](n)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(see [13,15]. Thus (2.3) and (2.4) are satisfied and A [member of] B([l.sub.p]) by Corollary 3.

Example 5. If [q.sub.n] = 1 and

[p.sub.n] = { 1 if n = [m.sup.2], m [member of] N, 0 otherwise,

then neither the conditions of Corollary 2 nor the conditions of Corollary 3 are satisfied but nevertheless (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) (see [2]).

2.3. The following corollary comes from Proposition A.

Corollary 4. Let [A.sub.1] = (N, [p.sub.1.sub.n], [q.sup.1.sub.n]) [member of] B[l.sub.p] and [A.sub.2] = (N, [p.sup.2.sub.n], [r.sup.1.sub.n]) [member of] B([l.sub.p]). n) [member of] B([l.sub.p]) and [A.sub.[alpha]][member of] = (N, [p.sub.2]

(i) Then also A = (N, ([p.sub.2] * [p.sub.1])n, [q.sup.1.sub.n]) [member of] B ([l.sub.p] and

n) [member of] B([l.sub.p]) and

[parallel]A[[parallel].sub.p] [less than or equal to] [parallel]A[[parallel].sub.2].sub.p] [parallel]A[[parallel].sub.1].sub.p].

(ii) In particular, if the sequences [p.sub.1] = ([p.sup.1.sub.n]) and [p.sub.2] = ([p.sup.2.sub.n]) are non-increasing and ([p.sup.1.sub.n]) is non-decreasing, then

[parallel]A[[parallel].sub.p] [less than or equal to] [q.sub.2].

Proof. As A is the product of matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then statement (i) is true by Proposition A and statement (ii) follows from (i) because by Corollary 1 we have for this particular case the inequalities [parallel]A[[parallel].sub.1].sub.p] [greater than or equal to].sub.q] and [parallel]A[[parallel].sub.2].sub.p] [less than or equal to] q.

3. SOME FAMILIES OF MATRICES BEING BOUNDED OPERATORS ON [l.sub.p]

We consider here some families of matrices

A = (N, [p.sup.[alpha].sub.n], [q.sub.n]),

where is [alpha] continuous or discrete parameter. These families of matrices have been studied in different papers (see, e.g., [9,13,14,16-20] on different levels of generality from the point of view of summability of sequences x = ([x.sub.n]).

Applying Corollaries 2-4, we find the sufficient conditions for [A.sub.[alpha]] [member of] B([l.sub.p]) but do not focus on proving estimates for the norms [parallel]A[[parallel].sub.[alpha]].sub.p].

Theorem 1. Let [A.sub.[alpha]] = (N, [p.sup.[alpha].sub.n], [q.sub.n]) be generalized Norlund matrices, where is a continuous parameter with values > 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where ([c.sup.[alpha].sub.n]) is either

(i) [c.sup.[alpha].sub.n] = [A.sup.[alpha].sub.n] = (n + [alpha]-1/1), n [member of][N.sub.0],

or

(ii) [c.sup.[alpha].sub.n] = [[alpha].sup.n]/[n!], n [member of] [N.sub.0].

If A = (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) and ([r.sub.n]) is almost non-decreasing, then also (N, [p.sup.[alpha].sub.n], [q.sub.n]) [member of] B([l.sub.p]) for any [alpha] > 0. In particular, if [r.sub.n] is non-decreasing, then in case (i) the inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] holds, where [[alpha]] is the integer part of[alpha]. More precisely, in this case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We prove the theorem first for the special case if [p.sub.0] = 1 and [p.sub.n] = 0 for any n [member of] N.

Lemma. Let us suppose that [A.sub.[alpha]] = (N, [c.sup.[alpha].sub.n], [q.sub.n]), where [alpha] is a continuous parameter with values [alpha] > 0, ([q.sub.n]) is almost non-decreasing, and [c.sup.[alpha].sub.n] is defined as in Theorem 1 in both cases (i) and (ii). Then [A.sub.[alpha]] [member of] B([l.sub.p]) for any [alpha] > 0.

In particular, if (qn) is non-decreasing, then

[parallel][[A.sub.[alpha][parallel].sub.p] [less than or equal to] [q.sup.[[alpha]]+1] ([alpha] > 0) (3.1)

in case (i). More precisely,

[parallel][[A.sub.[alpha][parallel].sub.p] [less than or equal to] [q.sup.[[alpha]] ([alpha] [member of]N). (3.2)

Proof. For case (ii) notice that the sequence ([c.sup.[alpha].sub.n]) is almost non-increasing and thus [A.sub.[alpha]] [member of] B([l.sub.p]) by Corollary 2.

In case (i) we choose some [alpha]>0 and show that [A.sub.[alpha]] [member of] B([l.sub.p]) and that (3.1) and (3.2) hold in our particular case. If [alpha] [less than or equal to] 1, then [c.sup.[alpha].sub.n]= [c.sup.[alpha]-1].sub.n] is non-increasing and our statement is true by Corollary 2.

If [alpha] > 1, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is increasing. We use the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(see, e.g., [19]). Taking [sigma]= 1, we can represent [A.sub.[[alpha]]] in the form of the product

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

The right side of equality (3.3) is a product of [[alpha]] matrices. As [A.sup.0.sub.n] = 1, each of these matrices is in B([l.sub.p]) by Corollary 2 and therefore [A.sub.[[alpha]] [member of] B([l.sub.p]) by Proposition A. In particular, if ([q.sub.n]) is non-decreasing, then each of the factors in the right side of equality (3.3) has a norm not greater than q by Corollary 1. As a result, we get the inequality

[parallel][[A.sub.[alpha][parallel].sub.p] [less than or equal to] [q.sup.[[alpha]]

in this particular case by Proposition A again. Thus, for [alpha] = [[alpha]] our statement is proved. For [alpha] > 1 in general we have the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As both factors in the right side of the last equality are in B([l.sub.p]) and the norm of the first of them is not greater than q, [A.sub.[alpha]] is in B([l.sub.p]), and also inequality (3.1) holds in the particular case by Proposition A.

Proof of Theorem 1. We have the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for any [alpha] > 0. where the right side is the product of matrices. As [r.sub.n] is almost non-decreasing, (N, [c.sup.[alpha].sub.n], [r.sub.n]) [member of] B([l.sub.p]), and also (3.1) and (3.2) hold in the particular case by Lemma. Thus our statement is true by Proposition A.

Example 6. If A = (N, [p.sub.n], [q.sub.n]) is defined as in Examples 1, 2, 3, or 5, then (N, [c.sup.[alpha].sub.n], [r.sub.n]) [member of] B([l.sub.p]) for any [alpha]> 0 by Theorem 1, because (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) and [r.sub.n] is non-decreasing in these cases.

Remark 2. The best-known special cases of the matrices (N, [c.sup.[alpha].sub.n], [r.sub.n]) given in Theorem 1 in case (i) are the Cesaro matrices (C, [alpha]), where [p.sub.n] = [[delta].sub.0n] and [q.sub.n] = 1. and the generalized Cesaro matrices (C, [alpha], [gamma]), where [p.sub.n] = [[delta].sub.0n] and [q.sub.n] = (n+[gamma] n). An example of case (ii) is given by Euler-Knopp matrices (E, [alpha]) with [p.sub.n] = [[delta].sub.0n] and [q.sub.n] = 1/n!.

Theorem 2. Consider the matrices [A.sub.[alpha]] = (N, [p.sup.[alpha].sup.n], [q.sup.n]) = (N, [p.sup.*[alpha].sup.n], [q.sup.n]) with the discrete parameter [alpha] [member of] N defined by the convolutions [p.sup.*1] = ([p.sup.n]) and ([p.sup.[alpha].sub.n]) = [p.sup.*[alpha]] = [p.sup.*[alpha].sub.n]]) = [p.sup.*1] [p.sup.*([alpha]-1]) for any [alpha]= 2,3, ...

If

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [L.sub.2](*) are slowly varying functions and [L.sub.2](*) is non-decreasing, or

(ii) ([p.sub.n]) is almost non-increasing and ([q.sub.n]) is almost non-decreasing, then [A.sub.[alpha]] [member of] B([l.sub.p]) for any [alpha] [member of] N. In particular, if ([p.sub.n]) is non-increasing and ([q.sub.n]) is non-decreasing, then [parallel][[A.sub.[alpha][parallel].sub.p] [less than or equal to] [q.sup.[alpha]].

Proof. In case (i) we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where [L.sup.[alpha].sub.1] (*) is also a slowly varying function (see [13]). Thus [A.sub.[alpha]] [member of] B([l.sub.p]) as was shown in Example 4.

In case (ii) we use the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is almost non-decreasing because ([q.sub.n]) is almost non-decreasing. As (N, [p.sub.n], [q.sub.n]) [member of] B([l.sub.p]) and (N, [p.sub.n]. [r.sup.[alpha].sub.n]) [member of] B([l.sub.p]) for any [alpha] [member of] N by Corollary 2, the relation [A.sub.[alpha]] [member of] B([l.sub.p]) and also the estimate of the norm [parallel][[A.sub.[alpha]][[parallel].sub.p] follow from Corollary 4 by induction.

Remark 3. We note that the matrices [A.sub.[alpha]] = (N, [p.sup.[alpha].sub.n]], [q.sub.n]), which satisfy the conditions of Theorems 1 or 2 and are therefore bounded operators on [l.sub.p], need not satisfy the conditions neither of Corollary 2 (Theorem A) nor of Corollary 3 (Theorem B). For example, if ([p.sub.n]) is an almost non-increasing sequence, then ([p.sup.[alpha].sub.n]) need not be almost non-increasing any more. Moreover, ([p.sup.[alpha].sub.n]) is non-decreasing for any [alpha] [greater than or equal to] 1 in case (i) of Theorem 1.

We finish our paper with an application of Corollary 3.

Theorem 3. Suppose that [A.sub.[alpha]] = (N, [p.sup.[alpha].sub.n], [q.sub.n]) ([alpha] > 0) are the same matrices as in Theorem 1 in case (i). If ([p.sub.n]) satisfies condition (2.4), ([q.sub.n]) is almost non-decreasing and satisfies the condition

(n+1)[q.sub.n] = O([Q.sub.n]),

then [A.sup.[alpha]] [member of] B([l.sub.p]) for any [alpha]> 0.

Proof. We apply Corollary 3 to the methods (N, [p.sup.[alpha].sub.n], [q.sub.n]) (instead of the methods (N, [p.sub.n], [q.sub.n])). We know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [P.sub.n][Q.sub.n] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see [18]). Thus condition (2.3) is satisfied.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and [A.sub.[alpha]] [member of] B([l.sub.p]) for any [alpha]> 0 by Corollary 3.

DOI: 10.3176/proc.2009.3.01

ACKNOWLEDGEMENT

This research was supported by the Estonian Science Foundation (grant No. 7033).

Received 11 December 2008, revised 21 January 2009, accepted 21 January 2009

REFERENCES

[1.] Borwein, D. Generalized Hausdorff and weighted mean matrices as bounded operators on l p. Math. Z., 1983, 183, 483-487.

[2.] Borwein, D. Norlund operators on [l.sup.p]. Canad. Math. Bull., 1993, 36, 8-14.

[3.] Borwein, D. Simple conditions for matrices to be bounded operators on [l.sup.p]. Canad. Math. Bull., 1998, 41, 10-14.

[4.] Borwein, D. Weighted mean operators on [l.sup.p]. Canad. Math. Bull., 2000, 43, 406-412.

[5.] Borwein, D. and Cass, F. P. Norlund matrices as bounded operators on [l.sup.p]. Arch. Math., 1984, 42, 464-469.

[6.] Borwein, D. and Gao, X. Generalized Hausdorff and weighted mean matrices as operators on [l.sup.p]. J. Math. Anal. Appl., 1993, 178, 517-528.

[7.] Borwein, D. and Gao, X. Matrix operators on [l.sup.p] to [l.sup.q]. Canad. Math. Bull., 1994, 37, 448-456.

[8.] Borwein, D. and Jakimovski, A. Matrix operators on [l.sup.p]. Rocky Mountain J. Math., 1979, 9, 463-477.

[9.] Cass, F. P. Convexity theorems for Norlund and strong Norlund summability. Math. Z., 1968, 112, 357-363.

[10.] Cass, F. P. and Kratz, W. Norlund and weighted mean matrices as operators on [l.sup.p]. Rocky Mountain J. Math., 1990, 20, 59-74.

[11.] Hardy, G. H. An inequality for Hausdorff means. J. London Math. Soc., 1943, 18, 46-50.

[12.] Jakimovski, A., Rhoades, B. E., and Tzimbalario, J. Hausdorff matrices as bounded operators over [l.sup.p]. Math. Z., 1974, 138, 173-181.

[13.] Kiesel, R. General Norlund transforms and power series methods. Math. Z., 1993, 214, 273-286.

[14.] Kiesel, R. On scales of summability methods. Math. Nachr., 1995, 176, 129-138.

[15.] Kiesel, R. The law of the iterated logarithm for certain power series and generalized Norlund methods. Math. Proc. Camb. Phil. Soc., 1996, 120, 735-753.

[16.] Kiesel, R. and Stadtmuller, U. Tauberian and convexity theorems for certain (N, p,q)-means. Can. J. Math., 1994, 46, 982- 994.

[17.] Sinha, R. Convexity theorem for (N, p,q) summability. Kyungpook Math. J., 1973, 13, 37-40.

[18.] Stadtmuller, U. and Tali, A. On some families of certain Norlund methods and power series methods. J. Math. Anal. Appl., 1999, 238, 44-66.

[19.] Stadtmuller, U. and Tali, A. Comparison of certain summability methods by speeds of convergence. Anal. Math., 2003, 29, 227-242.

[20.] Tali, A. Convexity conditions for families of summability methods. Tartu alik. Toimetised, 1993, 960, 117-138.

Ulrich Stadtmullera and Anne Tali (b) *

(a) Department of Number Theory and Probability Theory, Ulm University, 89069 Ulm, Germany

(b) Department of Mathematics, Tallinn University, Narva mnt. 25, 10120 Tallinn, Estonia

* Corresponding author, atali@tlu.ee