On the $K_r$ -core of complex sequences and the absolute equivalence of summability matrices

Mathematical and Computational Applications, Vol. 9, No. 3, pp. 409-416, 2004.

© Association for Scientific Research

ON THE Kr -CORE OF COMPLEX SEQUENCES AND THE ABSOLUTE EQUIVALENCE OF SUMMABILITY MATRICES

Celal Çakan* and Cafer Aydın**

*İnönü Üniversitesi Eğitim Fakültesi, 44069-Malatya, Turkey ccakan@inonu.edu.tr

**Kahramanmaraş Sütçü İmam Üniversitesi, Fen-Edebiyat Fakültesi, Kahramanmaraş, Turkey

Abstract- The Kr -core of a complex valued sequence has been introduced in [5]. In this paper, we have determined a class of matrices such that Kr - core (Ax) ⊆ Kr - core (x) hold for all x∈ l∞. Also; we have defined a new type of absolute equivalence, r - absolute equivalence, and characterized these type of matrices.

Key words- Sequence spaces, absolute equivalence, matrix transformations and core theorems.

1. INTRODUCTION

Let A be an infinite matrix of complex entries ank (n, k ∈ N, the set of natural numbers) and x = (xk) be a sequence of complex numbers. Then Ax = {(Ax)n} is called the A transform of x, if (Ax)n =

∑

xk for all x ∈ X, then we say A regularly transforms X into Y and write A ∈ (X, Y)reg. The matrix A ∈ (c, c)reg is said to be regular and the conditions of regularity are well-known, [4, pp. 4], where c is the space of all convergent complex sequences.

The regular matrices A and B are said to be absolutely equivalent on l∞, the space of all bounded complex sequences, [4, pp. 97] if lim (Ax - Bx) = 0, (i.e., Ax and Bx have the same limit or neither of them goes to a limit but their difference goes to zero). It is also well-known [4, pp. 105] that the regular matrices A and B are absolutely equivalent on l∞ if and only if

0

|

|

lim

∑

k

nk

n ank b .

Let us define, for any real number r, the matrix A^r = (a ) by _nk^r







>

+ ≤ +

=

. , 0

1 , 1

n k

n n k r a

k r

nk

In [3], it is shown that the matrix A^r is regular for 0< r <1 and it is stronger than the Cesáro matrix defined by







>

+ ≤

=

. , 0 1 ,

1

n k

n n k c_nk

C. Çakan and C. Aydın 410

The β–dual space of a sequence space X is defined by







 ∈

= ^a

∑

X

k k k k):

β (

and by XA we mean the set of sequences such that Ax ∈ X, i.e., }

: )

{(x Ax X

X_A = _k ∈ .

In [2], the new sequence space a is defined by _c^r c and it is shown that Ar

β = ) (a_c^r







 ∈









∞ +

<

 +



 





∆ +

∑

a ^k _k

k k

k

k ( 1) 1

: 1 )

( ,

where cs is the space of all convergent series. Note that if a sequence x ∈ a , then we _c^r write a -lim x exists. Also, the sequence space _c^r a_∞^r is introduced as (l )_∞ Ar and it is established that

∞)β = (a^r







 ∈







 +

∞ +

<

 +



 





∆ +

∑

: )

( a c

r and k r k

a a _{k k}

k k

k

k ,

where c0 is the space of all null sequences.

Let us write

∑

+ +

=

= ⁿ

k k

k r

r

n r x

x n A x t

0

) 1 1 (

) 1 ( ) (

and Hn be the least closed convex hull containing t_n^r,t_n^r₊₁,t_n^r₊₂,.... In [5], Kr-core of a complex sequence x is defined by the intersection of all Hn. Also, it is shown that

I

z x

r core x G z

K

∈

=

− ( ) ( )

for any x ∈ l∞, where G_x(z)={w∈C:|w−z|≤limsup_n |t_n^r(x)−z|}and C is the set of all complex numbers.

In the present paper, we have determined the necessary and sufficient conditions on a matrix A for which Kr-core (Ax) ⊆ Kr -core (x) for all x ∈ l∞. Also, we have

introduced a new type of absolute equivalence, r-absolute equivalence, and characterized the r-absolutely equivalent matrices.

2. THE INCLUSION THEOREM

Firstly, we shall quote some lemmas which will be useful to our proofs.

Lemma 2.1 [5, Lemma 2.1]. A ∈(l∞, a ) if and only if _c^r (2.1) ^{|| ||} =^sup

∑

k nk n

r a

A

(2.2) _{nk k}

n a~ =α

lim for each k, (2.3) lim

∑

k

k

n ank α ,

On The Kr -Core of Complex Sequences 411

where

a~ = nk

∑

=

+ +

n

j

jk k a n ₀(1 r ) 1

1 , (n, k ∈N ).

Lemma 2.2. A∈(a_∞^r,c) if and only if

(2.4) 0

1

lim 1 =

+ +

k nk

k a

r

k , for each n,

(2.5)

∑

k n nk

t |

|

sup ,

(2.6) _{nk k}

n t =α

lim for each k, (2.7)

∑

k

k

n |tnk | 0

lim α ,

where ( 1)

1 ) 1

1

1 ( ¹

1

,  +

 





− +

= +

 +



 





∆ +

= ⁺₊ k

r a r k a

r

t_nk a^nk_k ^nk_k ^nk_k .

Next lemma is a special case of the Corollary 5.5 of [2] for s = 1.

Lemma 2.3. A∈(a_c^r,c)_reg if and only if the conditions (2.5) and (2.6) holds for α k = 0 and

(2.8) , (n N)

1 ∈ ∈









+ _∈ cs

r a

N k k nk

(2.9)

∑

k nk

n t 1

lim .

Lemma 2.4. A∈(a^r_c, a^r_c)_reg if and only if the condition (2.8) of Lemma 2.3 holds and

(2.10)

∑

k nk

n

a~

sup

(2.11) lim∆~_nk =0

n a for each k,

(2.12)

∑

k

n a~nk 1

lim where

) 1 1 (

~ 1

) ~ 1 1 (

~ ~

1 1

,  +

 





− +

= +

 +



 





∆ +

=

∆ ⁺₊ k

r a r k a

r

a_nk a^nk_k ^nk_k ^nk_k .

Proof. Let x∈ and consider the equality a_c^r

C. Çakan and C. Aydın 412

(2.13)

∑ ∑ ∑ ∑

= =

= =

+ +

= + +

m

k

n

j

k jk n j

j

m

k

k jk

j r a x

x n a

n ₀ r _{0 0 0}(1 )

1 ) 1

1 1 (

1

=

∑

= m

k

k nkx d

0

, (m, n ∈ N) which yields for m→∞ that

) ( , ) ( ) )(

1 1 (

1

0

N n Dx Ax

n r ⁿ

n

j j

j = ∈

+

∑

=

where )D=(d_nk defined by







>

≤

≤ + +

=

∑

=

n).

(k ,

0

n) k (0 , ) 1 1 (

1

0 n

j

jk j nk

a n r

d

Furthermore, since the spaces a and c are linearly isomorphic (see [2]), we deduce _c^r from that A∈(a_c^r,a_c^r)_reg if and only if D∈(a_c^r,c)_reg. Therefore, the necessary and sufficient conditions are obtained from the Lemma 2.3 by replacing the entries of matrix A by those of the matrix D .

Lemma 2.5. A∈(a_∞^r,a_c^r) if and only if the conditions (2.4) and (2.10) hold and (2.14) _{nk k}

n ∆~a =α

lim for each k, (2.15) lim

∑

k

k

n ank α .

Proof. For x∈a_∞^r , by (2.13), one can easily see that A∈(a_∞^r,a_c^r) if and only if )

, (a c

D∈ _∞^r . Hence, the proof follows from Lemma 2.2.

Following is a Steinhauss type theorem.

Lemma 2.6. The classes (a_∞^r , a ) and_c^r (a^r_c, a^r_c)_reg are disjoint.

Proof. Suppose, if possible, there exists a matrix A belonging to the two classes.

Then, Lemma 2.4 and 2.5 implies that

∑

k

n | a~nk | 0

lim .

On The Kr -Core of Complex Sequences 413

But since

∑

∑

k nk k

nk a

a~ | ~ |,

~ 0

limn

∑

Now, we may give our main theorem.

Theorem 2.7. Let A∈(a_c^r, a_c^r)_reg. Then, Kr -core (Ax) ⊆ Kr -core (x) for all x ∈ l∞

if and only if

(2.16)

∑

k nk

n | a~ | 1

lim .

Proof (Necessity). Suppose that the condition (2. 16) does not hold. Then,

∑

k

n | a~nk | 1

lim .

The conditions (2.10)-(2.12) allow us to choose two strictly increasing sequences {ni} and {k(ni )} (i = 1, 2, …) of positive integers such that

∑ ∑

+

=

= ₋

− ∆ < ^{( )} ∆ > +

1 ) (

, )

( 0

,

1 1

2 1 1

~ |

| 4,

| 1

| ~ ⁱ

i

i i

i

n k

n k k

k n n

k

k

k

n a

a

and

∑

+

=

<

∆

1 ) (

, 4

| 1

| ~

i

n i

k k

k

an .

Now, let us define a sequence x = (xk) by ~ , ( ) 1 ( )

1

,k i i

n

k sign a k n k k n

x = ∆ i ₋ + ≤ < .

Then, since Ar is regular, limsup_kt_k^r(x)≤1. Therefore, K_r −core(x)⊆{w∈C:|w|≤1}.

C. Çakan and C. Aydın 414

Also,

⁼

∑

k k n r

ni Ax a i

t ( )| ~ _,

|

∑ ∑ ∑

+

=

=

=

∆

−

∆

−

∆

≥ ⁻

− ( ) 1

, )

( 0

, )

( ) (

, | | ~ | | ~ |

| ~ ¹

1 i

i i

i i

i i

n k k

k n n

k

k

k n n

k

n k k

k

n a a

a

1 4 1 4 1 2

1+ 1− − =

> .

Since A∈(a^r_c, a^r_c)_reg, it follows that {t_n^r_i(Ax)} is bounded and hence {t_n^r(Ax)} has a

subsequence whose a -limit cannot be in ^r_c {w∈C:|w|≤1}. This is a contradiction with the fact that Kr -core (Ax) ⊆ Kr -core (x). Thus, (2.16) must be hold.

(Sufficiency). Let w ∈ Kr -core (Ax). Then, for any given z ∈ C , we can write (2.17) |w-z| ≤ limsup|t_n^r(Ax) z|

n

− = a x_k z

k nk

n

∑

≤ limsup

∑

∑

k nk

k n

k nk

n

a z z

x a

= limsup a~ (x_k z)

k nk

n

∑

k

=

− | ) (

| sup

lim . Then, for any ε > 0 there exists an increasing sequence (ks) of positive integers such that, |t_k^r_s(x)−z|≤l+ε whenever ks ≥ k0. Hence, one can write

(2.18) a~ (x z) a~ (t_k^r(x) z)

k nk k

k

nk − =

∑

∑

= ~ ( ( ) ) ~ ( ( ) )

0 0

z x t a z

x t

a _k^r

k k

nk r

k k

k

nk − + ∆ −

∆

∑

∑

≤

∑ ∑

<

∆ +

+

∆

−

0

~ |

| ) (

~ |

|

| ) (

| sup

k

k k

nk nk

r k k

a l

a x

t

z ε .

Therefore, applying the operator limsupn to (2.18) and using the hypothesis with (2.17), we have

On The Kr -Core of Complex Sequences 415

|w-z| ≤ limsup a~ (x_k z)

k nk

n

∑

This means that w ∈ Kr-core(x) and the proof is completed.

Our next theorem is an application of the Lemma 2.6 and Theorem 2.7. In this theorem we consider the real bounded sequences. In that case, the Kr-core of a sequence x is the closed interval [liminf_kt_k^r(x), )limsup_kt_k^r(x ].

Theorem 2.8. Let B be a matrix in the class (a^r_c, a^r_c)_reg satisfying the condition (2.16). Then, there is no matrix A such that limsup_nt_n^r(Ax)≤liminf_nt_n^r(Bx) for all x ∈ l∞.

Proof. Suppose, if possible, there exists such a matrix A. Theorem 2.7 implies that

) ( sup lim ) ( sup

lim _nt_n^r Bx ≤ _kt_k^r x , and so

) ( sup lim ) ( sup

lim _nt_n^r Ax ≤ _kt_k^r x ,

whence A ∈ (a^r_c, a^r_c)_reg. By the Lemma 2.6, there exists a z ∈ l∞ such that )

( sup lim ) ( inf

lim _nt_n^r Az < _nt_n^r Az . On the other hand, since

) ( sup lim ) ( sup

lim _nt_n^r Ax ≤ _nt_n^r Bx for all x ∈ l∞, we have

) ( inf lim ) ( inf

lim _nt_n^r Bz ≤ _nt_n^r Az . Thus,

) ( inf lim ) ( sup lim ) ( inf

lim _nt_n^r Bz ≤ _nt_n^r Ax ≤ _nt_n^r Bz contradicts to the fact B ∈ (a^r_c, a_c^r)_reg . This proves the theorem.

3. R - ABSOLUTE EQUIVALENCE

In this section, we introduced and characterized r-absolutely equivalence matrices.

Definition 3.1. The matrix A, B ∈ (a^r_c, a^r_c)_reg is said to be r-absolutely equivalent on l∞ if limt_n^r(Ax− Bx)=0for all x ∈ l∞.

Theorem 3.2. The matrices A, B ∈ (a^r_c, a^r_c)_reg are r-absolutely equivalent on l∞

if and only if

(3.1) lim

∑

k nk nk

n a b .

C. Çakan and C. Aydın 416

Proof. (Necessity). Let A and B be r-absolutely equivalent on l∞. Then, clearly the matrix D = (dnk) defined by dnk = (ank - bnk) is in the class (l∞, a₀^r ). Therefore, the necessity of the condition (3.1) follows from a result of Lemma 2.1.

(Sufficiency). Let the condition (3.1) hold and x ∈ l∞. In this case, we have

|Ax – Bx| =

∑

k

k nk

nk b x

a ~ )

(~

≤ x

∑

k

nk

nk b

a ~ |

|~ ,

which by (3.1) implies the r-absolute equivalence of A and B.

4. ACKNOWLEDGEMENT

We would like to express our great thanks to Professor Feyzi BAŞAR, İnönü University, Faculty of Education, Department of Mathematics Malatya- Turkey and the referee for their valuable suggestions improving the paper.

REFERENCES

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2. C. Aydın, F. Başar, On the new sequence spaces which include the spaces c0 and c, Hokkaido Math. J., (to appear).

3. F .Başar, A note on the triangle limitation methods, Fırat Üniv. Fen & Müh. Bil.

Dergisi, 5 (1), 113-117, 1993.

4. R. G. Cooke, Infinite matrices and sequence spaces, Macmillan, New York, 1950.

5. C. Çakan, C. Aydın, Some results related to the cores of complex sequences and the sequence space a , Thai. J. Math., 2(1), 2004, (to appear). _c^r