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The [chi square] sequence spaces defined by a modulus.

[section]1. Introduction

Throughout w, [chi] and [LAMBDA] denote the classes of all, gai and analytic scalar valued single sequences, respectively.

We write [w.sup.2] for the set of all complex sequences ([x.sub.mn]), where m, n [member of] N, the set of positive integers. Then, [w.sup.2] is a linear space under the coordinate wise addition and scalar multiplication.

Some initial works on double sequence spaces is found in Bromwich [4]. Later on, they were investigated by Hardy [8], Moricz [12], Moricz and Rhoades [13], Basarir and Solankan [2], Tripathy [2[degrees]], Colak and Turkmenoglu [6], Turkmenoglu [22], and many others.

Let us define the following sets of double sequences:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t = ([t.sub.mn]) is the sequence of strictly positive reals [t.sub.mn] for all m, n [member of] N and p- [lim.sub.m,n[right arrow][infinity]]denotes the limit in the Pringsheim's sense. In the case [t.sub.mn] = 1 for all m, n [member of]N; [M.sub.u] (t), [C.sub.p](t), [C.sub.0p](t), [L.sub.u] (t), [C.sub.bp] (t) and [C.sub.0bp] (t) reduce to the sets [M.sub.u], [C.sub.p], [C.sub.0p], [L.sub.u], [C.sub.bp] and [C.sub.0bp], respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gokhan and Colak [27,28] have proved that [M.sub.u] (t) and [C.sub.p] (t), [C.sub.bp] (t) are complete paranormed spaces of double sequences and gave the [alpha]-, [beta]-, [gamma]- duals of the spaces [M.sub.u] (t) and [C.sub.bp] (t). Quite recently, in her PhD thesis, Zelter [29] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [30] have recently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesaro summable double sequences. Nextly, Mursaleen [31] and Mursaleen and Edely [32] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = ([x.sub.jk]) into one whose core is a subset of the M-core of x. More recently, Altay and Basar [33] have defined the spaces BS, BS (t), [CS.sub.p], [CS.sub.bp], [CS.sub.r] and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces [M.sub.u], [M.sub.u] (t), [C.sub.p], [C.sub.bp], [C.sub.r] and [L.sub.u], respectively, and also examined some properties of those sequence spaces and determined the a-duals of the spaces BS, BV, [CS.sub.bp] and the [beta](v)-duals of the spaces [CS.sub.bp] and [CS.sub.r] of double series. Quite recently Basar and Sever [34] have introduced the Banach space [L.sub.q] of double sequences corresponding to the well-known space [l.sub.q] of single sequences and examined some properties of the space [L.sub.q]. Quite recently Subramanian and Misra [35] have studied the space [[chi square].sub.M] (p, q, u) of double sequences and gave some inclusion relations.

Spaces are strongly summable sequences were discussed by Kuttner [42], Maddox [43] and others. The class of sequences which are strongly Cesaro summable with respect to a modulus was introduced by Maddox [11] as an extension of the definition of strongly Cesaro summable sequences. Connor 144 further extended this definition to a definition of strong A-summability with respect to a modulus where A = ([a.sub.n,k]) is a nonnegative regular matrix and established some connections between strong A-summability, strong A-summability with respect to a modulus, and A- statistical convergence. In [45] the notion of convergence of double sequences was presented by A. Pringsheim. Also, in [46] - [49] and [50] the four dimensional matrix transformation [(Ax).sub.k, l] = [[summation].sup.[infinity].sub.m=1] [[summation].sup.[infinity].sub.n=1] [a.sup.mn.sub.kl] [x.sub.mn] was studied extensively by Robison and Hamilton. In their work and throughout this paper, the four dimensional matrices and double sequences have real-valued entries unless specified otherwise. In this paper we extend a few results known in the literature for ordinary (single) sequence spaces to multiply sequence spaces. This will be accomplished by presenting the following sequence spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f is a modulus function and A is a nonnegative four dimensional matrix. Other implications, general properties and variations will also be presented.

We need the following inequality in the sequel of the paper. For a, b [greater than or equal to] 0 and 0 < p < 1, we have

[(a + b).sup.p] [less than or equal to] [a.sup.p] + [b.sup.p]. (1)

The double series [[summation].sup.[infinity].sub.m,n=1] [X.sub.mn] is called convergent if and only if the double sequence ([s.sub.mn]) is convergent, where [s.sub.mn] = [[summation].sup.m,n.sub.i,j=1] [x.sub.ij](m, n [member of] N) (see [1]).

A sequence x = ([x.sub.mn]) is said to be double analytic if [sup.sub.mn] [[absolute value of [x.sub.mn]].sup.l/m+n] < [infinity]. The vector space of all double analytic sequences will be denoted by [[LAMBDA].sup.2]. A sequence x = ([x.sub.mn]) is called double gai sequence if [((m + n)! [absolute value of [x.sub.mn]]).sup.1/m+l] [right arrow] 0 as m, n [right arrow] [infinity]. The double gai sequences will be denoted by [chi square]. Let [phi] = {all finite sequences}.

Consider a double sequence x = ([x.sub.ij]). The [(m, n).sup.th] section [x.sup.[m,n]] of the sequence is defined by [x.sup.[m,n]] = [[summation].sup.m,n.sub.i,j=0] [x.sub.ij][T.sub.ij] for all m, n [member of] N; where [T.sub.ij] denotes the double sequence whose only non zero term is a 1/[(i + j)!] in the [(i, j).sup.th] place for each i, j [member of] N.

An FK-space or a metric space X is said to have AK property if ([T.sub.mn]) is a Schauder basis for X. Or equivalently [x.sup.[m,n]] [right arrow] x.

An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = ([x.sub.k]) [right arrow] ([x.sub.mn])(m, n [member of] N) are also continuous.

Orlicz [16] used the idea of Orlicz function to construct the space ([L.sup.M]). Lindenstrauss and Tzafriri [10] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space [l.sub.M] contains a subspace isomorphic to [l.sub.p] (1 [less than or equal to ] p < [infinity]). subsequently, different classes of sequence spaces were defined by Parashar and Choudhary [17], Mursaleen et al. [14], Bektas and Altin [3], Tripathy et al.[ 21], Rao and Subramanian [18], and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [9].

Recalling [16] and [9], an Orlicz function is a function M : [0, [infinity]) [right arrow] [0, [infinity]) which is continuous, non-decreasing, and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) [right arrow] [infinity] as x [right arrow] [infinity]. If convexity of Orlicz function M is replaced by subadditivity of M, then this function is called modulus function, defined by Nakano [15] and further discussed by Ruckle [19] and Maddox [11] and many others.

An Orlicz function M is said to satisfy the [[DELTA].sub.2]-condition for all values of u if there exists a constant K > 0 such that M (2u) [less than or equal to] KM (u) (u [greater than or equal to] 0). The [[DELTA].sub.2] -condition is equivalent to M (tu) [less than or equal to] KlM (u), for all values of u and for l > 1.

Lindenstrauss and Tzafriri [10] used the idea of Orlicz function to construct Orlicz sequence space

[l.sub.M] = {x [member of] w : [[infinity].summation over (k=1)] M ([[absolute value of [x.sub.k]]/[rho]]) < [infinity], for some [rho] > 0}.

The space [l.sub.M] with the norm

[parallel]x[parallel] = inf{[rho] > 0: [[infinity].summation over (k=1)] M ([[absolute value of [x.sub.k]]/[rho]])[less than or equal to ] 1}.

becomes a Banach space which is called an Orlicz sequence space. For M (t) = [t.sup.p] (1 [less than or equal to ] p < [infinity]), the spaces [l.sub.M] coincide with the classical sequence space [l.sub.p].

If X is a sequence space, we give the following definitions:

(i) X' = the continuous dual of X,

(ii) [X.sup.[alpha]] = {a = ([a.sub.mn]) : [[infinity].summation over (m,n=1)][absolute value of [a.sub.mn][x.sub.mn]] < [infinity], for each x [member of] X } ,

(iii) [X.sup.[beta]] = { a = ([a.sub.mn]) : [[infinity].summation over (k=1)] [a.sub.mn][x.sub.mn] is convegent, for each x [member of] X } ,

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

(v) Let X be an FK-space [contains] [phi]; then [X.sup.f] = {f([[T.sub.mn]) : f [member of] X' },

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

[X.sup.[alpha]], [X.sup.[beta]], [X.sup.[gamma]] are called [alpha]- (or Kothe-Toeplitz) dual of X, [beta]- (or generalized-Kothe-Toeplitz) dual of X, [gamma]- dual of X, [delta]- dual of X respectively. [X.sup.[alpha]] is defined by Gupta and Kamptan [24]. It is clear that [x.sup.[alpha]] [subset] [X.sup.[beta]] and [X.sup.[alpha]] [subset] [X.sup.[gamma]], but [X.sup.[alpha] [subset] [X.sup.[gamma]] does not hold, since the sequence of partial sums of a double convergent series need not to be bounded.

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [36] as follows

Z ([DELTA]) = {x = ([x.sub.k]) [member of] w : ([DELTA][x.sub.k]) [member of] Z},

for Z = c, [c.sub.0] and [l.sub.[infinity]], where [DELTA][x.sub.k] = [x.sub.k] - [x.sub.k+1] for all k [member of] N. Here w, c, [c.sub.0] and [l.sub.[infinity]] denote the classes of all, convergent, null and bounded sclar valued single sequences respectively. The above spaces are Banach spaces normed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by

Z ([DELTA]) = {x = ([x.sub.mn]) [member of] [w.sup.2] : ([DELTA][x.sub.mn]) [member of] Z},

where Z = [[LAMBDA].sup.2], [chi square] and [DELTA]([x.sub.mn] = ([x.sub.mn] - [x.sub.mn+1]) - ([x.sub.m+1n] - [x.sub.m+1n+1] = [x.sub.mn] - [x.sub.mn+1] - [x.sub.mn+1n] + [x.sub.m+1n+1] for all m, n [member of] N.

[section]2. Definitions and preliminaries

Throughout the article [w.sup.2] denotes the spaces of all sequences. [[chi square].sub.M] and [[LAMBDA].sup.2.sub.M] denote the Pringscheims sense of double Orlicz space of gai sequences and Pringscheims sense of double Orlicz space of bounded sequences respecctively.

Definition 2.1. A modulus function was introduced by Nakano [15]. We recall that a modulus f is a function from [0, [infinity]) [right arrow] [0, [infinity]), such that

(i) f (x) = 0 if and only if x = 0,

(ii) f (x + y) [less than or equal to] f (x) + f (y), for all x [greater than or equal to] 0, y [greater than or equal to] 0,

(iii) f is increasing,

(iv) f is continuous from the right at 0. Since [absolute value of (f(x) - f (y))] [less than or equal to] f ([absolute value of (x - y)]), it follows from condition,

(v) that f is continuous on [0, [infinity]).

Definition 2.2. Let p, q be semi norms on a vector space X. Then p is said to be stronger that q if whenever ([x.sub.mn]) is a sequence such that p ([x.sub.mn]) [right arrow] 0, then also q ([x.sub.mn]) [right arrow] 0. If each is stronger than the others, the p and q are said to be equivalent.

Lemma 2.1. Let p and q be semi norms on a linear space X. Then p is stronger than q if and only if there exists a constant M such that q(x) [less than or equal to] Mp(x) for all x [member of] X.

Definition 2.3. A sequence space E is said to be solid or normal if ([[alpha].sub.mn][x.sub.mn]) [member of] E whenever ([x.sub.mn]) [member of] E and for all sequences of scalars ([[alpha].sub.mn]) with [absolute value of [[alpha].sub.mn]] [less than or equal to] 1, for all m, n [member of] N.

Definition 2.4. A sequence space E is said to be monotone if it contains the canonical pre-images of all its step spaces.

Remark 2.1. From the two above definitions it is clear that a sequence space E is solid implies that E is monotone.

Definition 2.5. A sequence E is said to be convergence free if ([y.sub.mn]) [member of] E whenever ([x.sub.mn]) [member of] E and [x.sub.mn] = 0 implies that [y.sub.mn] = 0.

By the gai of a double sequence we mean the gai on the Pringsheim sense that is, a double sequence x = ([x.sub.mn]) has Pringsheim limit 0 (denoted by P-lim x=0) such that ((m + n)! [absolute value of [x.sub.mn]])[1/[m+n]] = 0, whenever m, n [member of] N. We shall denote the space of all P- gai sequences by [chi square]. The double sequence x is analytic if there exists a positive number M such that [[absolute value of [x.sub.jk]].sup.1/j+k] < M for all j and k. We will denote the set of all analytic double sequences by [[LAMBDA].sup.2].

Throughout this paper we shall examine our sequence spaces using the following type of transformation:

Defintion 2.6. Let A = ([a.sup.mn.sub.k,l]) denote a four dimensional summability method that maps the complex double sequences x into the double sequence Ax where the k, i- th term to Ax is as follows:

[(Ax).sub.kl] = [[infinity].summation over (m=1)] [[infinity].summation over (n=1)] [a.sup.mn.sub.kl][x.sub.mn],

such transformation is said to be nonnegative if [a.sup.mn.sub.kl] is nonnegative.

The notion of regularity for two dimensional matrix transformations was presented by Silverman and Toeplitz and [51] and [52] respectively. Following Silverman and Toeplitz, Robison and Hamilton presented the following four dimensional analog of regularity for double sequences in whcih they both added an adiditional assumption of boundedness. This assumption was made because a double sequence which is P-convergent is not necessarily bounded.

Definition2.7. The four dimensional matrix A is said to be RH-regular if it maps every bounded P-gai sequence into a P-gai sequnece with the same P-limit.

In addition to this definition, Robison and Hamilton also presented the following SilvermanToeplitz type multidimensional characterization of regularity in [50] and [46]:

Theorem 2.1. The four dimensional matrix A is RH-regular if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[RH.sub.5] : [[infinity].summation over (m=1)] [[infinity].summation over (m=1)][a.sup.mn.sub.kl] is P-convergent and

[RH.sub.6] : there exist positive numbers A and B such that [summation over (m, n>B)] [absolute value of [a.sup.mn.sub.kl]] < A.

Definition 2.8. A double sequence ([x.sub.mn]) of complex numbers is said to be strongly A-summable to 0, if P-[lim.sub.k], l[[summation].sub.m, n] [a.sup.mn.sub.kl] [((m + n)! [absolute value of ([X.sub.mn] - 0)]).sup.1/m+n] = 0.

Let [sigma] be a one-one mapping of the set of positive integers into itself such that [[sigma].sup.m](n) = [sigma]([[sigma].sup.m-1] (n)), m = 1, 2, 3,.... A continuous linear functional [phi] on [[LAMBDA].sup.2] is said to be an invariant mean or a [sigma]-mean if and only if

(i) [phi](x) [greater than or equal to] 0 when the sequence x = ([x.sub.mn]) has [x.sub.mn] [greater than or equal to] 0 for all m, n.

(ii) [phi](e) = 1 where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) [phi]({[x.sub.[sigma](m),[sigma](n)]}) = [phi]({[x.sub.mn]}) for all x [member of] [[LAMBDA].sup.2.]

For certain kinds of mappings [sigma], every invariant mean [phi] extends the limit functional on the space C of all real convergent sequences in the sense that [phi] (x)=lim x for all x [member of]C consequently C[subset][V.sub.[sigma]], where [V.sub.[sigma]] is the set of double analytic sequences all of those [sigma]- means are equal.

If x = ([x.sub.mn]), set Tx = [(Tx.sup.)1/m+n] = ([x.sub.[sigma](m), [sigma]{n)]). It can be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

We say that a double analytic sequence x = ([x.sub.mn]) is [sigma]- convergent if and only if x [member of] [V.sub.[sigma]].

Definition 2.9. A double analytic sequence x = ([x.sub.mn]) of real numbers it said to be [sigma]-convergent to zero provided that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this case we write [[sigma].sub.2]-lim x=0. We shall also denoted the set of all double [sigma]-convergent sequences by [V.sup.2.sub.[sigma]]. Clearly [V.sup.2.sub.[sigma]] [subset] [[LAMBDA].sup.2].

One can see that in contrast to the case for single sequences, a P-convergent double sequence need not be [sigma]-convergent. But, it is easy to see that every bounded P-convergent double sequence is convergent. In addition, if we let [sigma](m) = m + 1 and [sigma] (n) = n + 1 in then [sigma]-convergence of double sequences reduces to the almost convergence of double sequences.

The following definition is a combination of strongly A-summable to zero, modulus function and [sigma]-convergent.

Definition2.10. Let f be a modulus, A = ([a.sup.mn.sub.kl]) be a nonnegative RH-regular summability matrix method and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now define the following sequence spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If f (x) = x then the sequence spaces defined above reduce to the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Some well-known spaces are defined by specializing A and f. For example, if A = (C, 1, 1) the sequence spaces defined above reduces to [chi square] (f) and [[LAMBDA].sup.2] (f) respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a final illustration, let A = (C, 1, 1) and f (x) = x, we obtain the following spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]3. Main results

In this section we shall establish some general properties for the above sequence spaces.

Theorem 3.1. [chi square] (A, f) and [[LAMBDA].sup.2] (A, f) are linear spaces over the complex filed C.

Proof. We shall establish the linearity [chi square] (A, f) only. The other cases can be treated in a similar manner. Let x and y be elements in [chi square] (A, f). For [lambda] and [mu] in C there exist integers [M.sub.[lambda]] and [N.sub.[mu]] such that [absolute value of [lambda]] < [M.sub.[lambda]] and [absolute value of [mu]] < [N.sub.[mu]]. From the conditions (ii) and (iii) of Definition 2.1, we granted the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all k and l. Since x and y are [chi square] (A, f), we have [lambda]x + [mu]y [member of] [chi square] (A, f). Thus [chi square] (A, f) is a linear space. This completes the proof.

Theorem 3.2. [chi square] (A, f) is a complete linear topological spaces with the the paranorm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. For each x [member of] [chi square] (A, f), g ([theta]) is exists. Clearly g ([theta]) = 0, g (-x) = g (x), and g (x + y) [less than or equal to] g (x)+g(y). We now show that the scalar multiplication is continuous. Now observe the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the integer part of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In addition observe that g(x) and [lambda] approahces 0 implies g ([lambda]x) approaches 0. For fixed [lambda], if x approaches 0 then g ([lambda]x) approaches 0. We now show that for fixed x, [lambda] approaches 0 implies g ([lambda]x) approaches 0. Let x [member of] [chi square] (A, f), thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [epsilon] > 0 and choose N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

For each (k, l) and by the continuity of f as [lambda] [right arrow] 0 we have the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all (k, l). Thus g ([lambda]x) [right arrow] 0 as [lambda] [right arrow] 0. Therefore [chi square] (A, f) is a paranormed linear topological space.

Now let us show that [chi square] (A, f) is complete with respect to its paranorm topologies. Let ([x.sup.s.sub.mn]) be a cauchy sequence in [chi square] (A, f). Then, we write g ([x.sup.s] - [x.sup.t]) [right arrow] 0 as s, t [right arrow] [infinity], to mean, as s, t [right arrow] [infinity] for all (k, l),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Thus for each fixed m and n as s, t [right arrow] [infinity]. We are granted

f ((m + n)! [absolute value of ([x.sup.s.sub.mn] - [x.sup.t.sub.mn])]) [right arrow] 0

and so ([x.sup.s.sub.mn]) is a cauchy sequence in C for each fixed m and n. Since C is complete as s [right arrow] [infinity] we have [x.sup.s.sub.mn] [right arrow] [x.sub.mn] for each (mn). Now from Definition 2.9, we have for [epsilon] > 0 there exists a natural numbers N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since M is arbitrary, by letting M [right arrow] [infinity] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all (k, l). Thus g ([x.sup.s] - x) [right arrow] 0 as s [right arrow] [infinity]. Also ([x.sup.s]) being a sequence in [chi square] (A, f) be definition of [chi square] (A, f) for each s with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as (k, l) [right arrow] [infinity] thus x [member of] [chi square] (A, f).

This completes the proof.

Theorem 3.3. Let A = ([a.sup.mn.sub.kl]) be nonnegative matrix such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and let f be a modulus, then [chi square] ( A, f) [subset] [[LAMBDA].sup.2] (A, f).

Proof. Let x [member of] [chi square] (A, f). Then by Definition 2.1 of (ii) and (iii) of the modulus function we granted the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There exists an integer [N.sub.p] such that [absolute value of 0] [less than or equal to] [N.sub.p]. Thus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and x [member of] [chi square] (A, f), we are granted x [member of] [[LAMBDA].sup.2] (A, f) and this completes the proof.

Theorem 3.4. Let A = ([a.sup.mn.sub.kl]) be nonnegative matrix such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and let f be a modulus, then [[LAMBDA].sup.2] (A) [subset] [[LAMBDA].sup.2] (A, f).

Proof. Let x [member of] [[LAMBDA].sup.2] (A) , so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [epsilon] > 0 and choose [delta] with 0 < [delta] < 1 such that f (t) < [epsilon] for 0 [less than or equal to] t [less than or equal to] [delta]. Consider,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

For

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [t] denoted the integer part of t and from conditions (ii) and (iii) of Definition 2.1, modulus function we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which together with inequality (7) yields the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and x [member of] [[LAMBDA].sup.2] (A), we are granted that x [member of] [[LAMBDA].sup.2] (A, f) and this completes the proof.

Definition 3.1. Let f be modulus be [a.sup.mn.sub.kl] nonnegative RH-regular summability matrix method. Let p = ([p.sub.mn]) be a sequence of positive real numbers with 0 < [p.sub.mn] < sup [p.sub.mn] = G and D = max (1, [2.sup.G-1]). Then for [a.sub.mn], [b.sub.mn] [member of] N, the set of comples numbers for all m, n [member of] N, we have

[[absolute value of ([a.sub.mn] + [b.sub.mn])].sup.1/m+n][less than or equal to ] D{[[absolute value of [[a.sub.mn]].sup.1/m+n] + [[absolute value of [b.sub.mn]].sup.1/m+n]}.

Let (X, q) be a semi normed space over the field C of complex numbers with the semi norm q. We define the following sequence spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.5. Let [f.sub.1] and [f.sub.2] be two modulus. Then [chi square] (A, [f.sub.1], p, q)[intersection][chi square] (A, [f.sub.2], p, q) [subset or equal to] [chi square] (A, [f.sub.1] + [f.sub.2], p, q).

Proof. The proof is easy so omitted.

Remark 3.1. Let f be a modulus [q.sub.1] and [q.sub.2] be two seminorm on X, we have

(i) [chi square] (A, f, p, [q.sub.1])[intersection] [chi square] (A, f, p, [q.sub.2]) [subset or equal to] [chi square] (A, f, p, [q.sub.1] + [q.sub.2]),

(ii) If [q.sub.1] is stronger than [q.sub.2] then [chi square] (A, f, p,[q.sub.1])[subset or equal to] [chi square] (A, f, p, [q.sub.2]),

(iii) If [q.sub.1] is equivalent to [q.sub.2] then [chi square] (A, f, p, [q.sub.1]) = [chi square] (A, f, p, [q.sub.2]).

Theorem 3.6. Let 0 [less than or equal to] [p.sub.mn] [less than or equal to] [r.sub.mn] for all m, n [member of] N and let {[q.sub.mn]/[p.sub.mn]} be bounded. Then [chi square] (A, f, r, q) [subset] [chi square] (A, f, p, q).

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

we have [[gamma].sub.mn] = [p.sub.mn]/[r.sub.mn]. Since [p.sub.mn] [less than or equal to] [r.sub.mn], we have 0 [less than or equal to] [[gamma].sub.mn] [less than or equal to] 1. Let 0 < [gamma] < [[gamma].sub.mn] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Hence

x [member of] [chi square] (A, f, p, q). (12)

From (8) and (12) we get x [member of] [chi square] (A, f, r, q) [subset] x [member of] [chi square] (A, f, p, q) .

Theorem 3.7. The space x [member of] [chi square] (A, f, p, q) is solid and a such are monotone.

Proof. Let ([x.sub.mn]) [member of] x [member of] [chi square] (A, f, p, q) and ([[alpha].sub.mn]) be a sequence of scalars such that, [absolute value of [[alpha].sub.mn]] [less than or equal to] 1 for all m, n [member of] N. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all m, n [member of] N. This completes the proof.

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N. Subramanian, Department of Mathematics, SASTRA University, Tanjore, 613402, India

U. K. Misra, Department of Mathematics, Berhampur University, Berhampur, 760007, Orissa, India

Vladimir Rakocevic, Faculty of Mathematics and Sciences,University of Nis, Visegradska 33, 18000 Nis Serbia

E-mail: nsmaths@yahoo.com umakanta_misra@yahoo.com vrakoc@bankerinter.net
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