CESARO SEQUENCE SPACES *

Lee Peng Yee

(received 19 May, 1984)

1. Introduction

The study of sequence spaces has been of great interest recently. A

number of books have been published in this area over the last few years

(see, for example, [6], [19] and [25]). In this lecture, we shall intro­

duce two classes of sequence spaces, one of which is of a nonabsolute type

and another of an absolute type. By absolute we mean that if a sequence

x = {x^} belongs to a given space, so does its absolute value |x| ={|x^|> .

Otherwise the space is said to be nonabsolute. We shall characterize among

others the dual spaces of these sequence spaces and give two applications

in Sections 3 and 5 respectively.

For convenience, we list some of the well-known spaces as follows.

o = the space of all convergent sequences,

= the space of all null sequences,

£ = the space of all sequences x = {x, } such thatP *

( f |x, < ♦ °« where 1 < p < « ,k=1

= the space of all bounded sequences,

bv = the space of all sequences x = {x^} such that

bvo = bv 0 .

For suitable norms, the above are all complete normed linear spaces, i.e.

Banach spaces [6] .

* Invited address delivered at the 19th Mathematics Colloquium, Victoria University, May 1984.

Math. Chronicle 13(1984) 29-45

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In what follows, we assume that the sequence spaces considered are

normed and complete. Also, we write x for {xfc} and y for {y .

2. Cesaro sequence spaces of a nonabsolute type

Let A be an infinite matrix. We define

- {x; Ax i a} .

This sequence space has been studied by many; see, for example, [24] .

In general, let Y be a given sequence space. We define

X = {x; Ax € Y} .

In what follows, we assume that the mapping of A from X to Y is one-

one and onto. In particular, when A is a Cesaro matrix C where

C =

11 / 21/3

1/21/3 1/3

and Y = I for 1 < p 5 we call X Cesaro sequence space of a non-

absolute type [17] and denote it by X . In other words, x € X for

1 < p < oc if and only if

1 Ip

and similarly for p

This class of sequence spaces has some undesirable properties. First,

it does not necessarily contain all finite sequences. For example,

e l = (1,0,0,...) ̂X j ; however X is not empty since

{(fe+1)’1 -k 1}ji;>1 * • Secondly, it is not solid. A set X is said to

be solid if x € X and \y\ 5 |x| imply that y € X . Note that if X

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kis solid then X is also absolute. Now let x = {x̂ ,} and x^ = (-1) .

Then x € X but Ixl I X for 1 < p < « . That is, X is nonabsolute p 1 1 r p r P

for 1 < p < « , indeed also for p = 1 and °° , therefore X cannot

be solid. We shall see that many results remain valid without the above

two properties which are usually assumed in the study of sequence spaces.

One area of study in sequence spaces is matrix transformations. Let

T - be an infinite matrix mapping a sequence space X into another

sequence space Z . That is, y = Tx i Z whenever x i X where

oo

yn = J , tn k xk ' k=1

The problem is to find necessary and sufficient conditions on T such

that T maps X into Z . A necessary condition is that for each n

and each x € X the series

\ t ^ x ^ must converge. k= 1

In other words, belongs to the 8-dual of X for each n .

We define two kinds of dual spaces here. The 6-dual of X , denoted by g

X , is the space of all sequences y such that for every x € X

I x,y. converges. fc-1

The a-dual of X , denoted by Xa , is the space of all sequences y

such that for every x i X

001 k, y A converges.

k=l K *

If the space X is solid, then X01 = For example, Ip = Up = I^

where 1 i p 5 » and 1/p + 1/q = 1 . However bva = whereas bv& - os .

Here as denotes the space of all sequences x such that

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£ x, converges.fc-1

In order to study matrix transformations on , the first natural

question to ask is what is the 3-dual of X^ .

s •Suppose y € X where 1 < p < « . Consider

n -1

£ (F X. xi > (kl"k - W > *<» V K i 1fc=l i=l i=l

where

n

fc*l *

k[yk when 1 - k - n - 1

n y ~when k - n ,

when k > n

andF .1 • t = l

Let B = (^n^) • Then for every x € X and s = {s^} € , we have

Bs i a . It is well-known [13] that S maps fc into c if and only if

Ziff? exists for each k , and

{( I IK v \q ) 1/c?; " - 1 > < +SUP XK L \V k k= 1 n*

where 1/p + 1/q = 1 . Note that the first condition is trivially satis­

fied. The second condition gives the following [17] .

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Theorem 1. The Q-dual of for 1 < p < « is the space of all sequences

y such that

n-1 q asup {( I IH y k -«fc+1)l + \nvn \ ) i M -i > < + “ •

k-1

We remark that similar results hold for p = 1 and « .

Alternatively, we may split the condition into two as follows:

(1) sup { \nyn \; n » 1 > < + - ;

(2) ( I \k (yk -yk+i)\q ) 1/q < * m •

Using condition (1), we see that

oo OO

^ x k^k = ̂ sk ̂ k k=l K K fc=l K K

where s^ is defined as above and t^ = -^ +i) • In other words, the

left hand side converges if and only if the right hand side also converges.

Since it is easy to show that condition (1) is necessary, we have an alter­

native proof of Theorem 1 .

In fact, the first proof is rather general. We can easily extend it

to a general case. For a given sequence x , we write

XN = ( x l t x 2 , . . . , X N , 0 , . . . ) .

Then we say a sequence space X has the AK property if for each x € X N N

we have x € X and ||x -x|| — ► 0 as N — *• « . Further, given a norm

in j; , we define an associate norm as follows:

S“P { I I x, y. | ; l|x|! < 1 } . k»1 K K

Hence we can prove [9]

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Theorem 2. Let A be an infinite matrix which is one-one and onto and

Y a sequence space having the AK property. Let X = (i ; i4i (7} . Then

the 8-dual of X is the space of all sequences y such that yA~l

exists and

sup { \\yNA 1 ||y ;N > 1} < + <» .

Obviously, Theorem 1 is a special case of Theorem 2. For other special

cases, see, for example, [16] . We remark that given a sequence space Xg

which is nonabsolute, an appropriate norm in X is given by

Hi,||B = sup { ||/||* ;tf>l}

and not l!j/||̂. . The latter is suitable only for sequence spaces that are

absolute.

3. Matrix transformations

We state a theorem due to Zeller.

Theorem 3. Let X} Y and Z be sequence spaces with Y having the

AK property arid A an infinite matrix such that A : X -+Y is one-one

and onto. Then T = (t m a p s X into Z if and only if

(i) ^nk^k^il 6 for each n ,

(ii) TA 1 maps Y into Z .

In view of Theorem 2, we can replace condition (i) above by those

given in Theorem 2. In particular, we have

Theorem 4. An infinite matrix T = (.t^) maps X^ into c for

1 < p < « if and only if

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(i) sup{\ktn^\; k > ! } < + “ for each n ,

< + » for each n ,

(iii) lim (t ^ -t ^+1) exists for each k , n-+ • *

f W o p f J t ! ) < * - •

Similarly, we can state the result for p = 1 and • . Note that as

long as we can characterize TC-1 : I — *-Z , which is normally easier, then

we can characterize T :X^— *Z . Hence we may obtain many other results.

The proof of Theorem 3 depends on the following associative property:

T(A~ly) = {.TA~l)y for all y € Y , which follows from the AK property

of Y by a theorem of Zeller [26; Theorem 17. vii]. We remark that Jaki-

movski and Livne [5] proved a similar result. There they assume that Y

satisfies a more general condition than AK .

In the same way, we can characterize T : Z — *X where Z has the AK

property and X is defined as in Theorem 3. Other extensions are indicated

in [9] . For a list of results on the characterization of infinite matrices

from some well-known sequence spaces into others, see [22] .

4. Cesaro sequence spaces of an absolute type

Now we consider the absolute version. Let A be an infinite matrix

and Y a sequence space. We consider

X = {x; 4|x| € Y},

In particular, when A is a Cesaro matrix C and Y = I for 1 < p < « ,

we call X Cesaro sequence space of an absolute type and denote it by

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ceSp . In other words, x € oesp ^or 1 < p < °° if and only if

P1**1 )

_ n=l n k-1

and similarly for p = °» . We do not consider the case when p = 1 be­

cause the space is trivial. The space aesp was first defined by Shiue

[20], its a-dual given by Jagers [4] for 1 < p < « , and by Ng and Lee

[15] for p = « . The version we shall present here is due to Lim [12] .

The first problem is to find the a-dual of . We remark that

the proof that works for the nonabsolute case does not go through here.

We observe that

X = J , j l*il)<»l»„l>K . — 1 K . — x. " I s — 1 I s — X

n-1= y s, t7 + taiZ-

k=l k

1 kwhere sfe = ^ J |xj and = fc( |«/fc | - |t^+11) •

i=l

If the tail tends to zero and £ = {*£,} € £. with 1/p + 1/q = 1 , thenCL ^

y € ees . But the converse does not hold because the seriesP

00 oo

1 \x i , y k I = I s k t k

k=1 k=l K K

if exists, converges for some s = {s, } € £ only. Hence a differentK p

approach is required.

Suppose y € ces^01 . Using functional analytic method [26; Theorem

17. iii], we can show that for every x € eesp

00

1 kfc&J - tflW1 • fc=i *

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Here the norm is defined to be

’ I (if j l*J>P] . fe=1 K t=l J

U p

It is easy to show that y € cQ and the tail as given above tends to zero.

Then we obtain

l ( r I l*vl)(*(li/J-Ktll)) 5 « I (jf n ^ | ) P k= l t=l 1 K K 1 - k-i K t=l 1

1/P

or alternatively,

I sk tk < M ( ̂ J sfeP )1/P

fc*l fc=l

-1In order to say t - {t^} 6 £ , we require = t £ . In other words,

we want to choose x € ces such thatP

jf I l*f lt=l

or equivalently,

.1 W - *,(l»*l-li/^l l)<rx - 1

-1

q-1

This is possible only when k ? (|y^\- l#£+1l) is non-negative and in­

creasing in k . Therefore we have proved

Lemma 5. The space contains all y such that y € cQ and

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I [ k(!j/fel-lyfe+1l) J q < + ~ k= 1

where fc^(\y^|- |y^+1I) non-negative and increasing; and its solid

hull.

Rewriting the above lemma, we have

Theorem 6. The a-dual of for 1 < p < » is the space of all

sequences y such that y € cQ and

oo

~ * I I V / * *where y denotes the infimum of all y < |z/1 with k

being non-negative and increasing.

Again, let X = {x; d|x| € Y} . Lim [12] considered the cases for

other special A and Y ; in particular, when A = C , Cesaro matrix,

and Y = wQ where wQ = {x; C |x| € <?0) .

5. Cesaro function spaces

There is a function version of Cesaro sequence spaces [21] . Cesaro

function space, denoted by CES^ for 1 < p < » , is defined to be the

set of all real-valued measurable functions defined on (0,<») such that

converges.

In other words, f € if and only if T\f\ 6 (0,<*>) where

'0

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Obviously, the operator T plays the role of matrix A or C as in the

sequence version.

Given a function space X with functions defined on (0,-) , the

Kothe dual of X is the space of all real-valued measurable functions g

defined on (0,«) such that

f f{x)g{x)dx exists for every f £ X .■'o

Unfortunately, it is difficult to express the inverse of T precisely.

Hence the technique used in the sequence version to find the dual does not

work here. However it is still possible to find the dual of CES' . We do

it by first of all converting the problem in function spaces to one in

sequence spaces, and then solve the problem in sequence spaces.

We observe that for n 5 x 5 n +1

1 n 1 x 1 ,n+1 9 rn+1

Sa ~ J0 1 /(t)| dt - n I ~ nTT (j

In other words, the integral

, x

1 0

if and only if the series

1 t x

f (— J |/(t) \dt) P dx converges

- j » pI ( ~ I ) converges

where

n=l n k=l

k

Ik-1

sk = | | f(t) | dt .

That is, s = {s, } € aesk p

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Next, we write for (m+1) 1 S x < m 1

, ,l/(m+l) .1/(m+1) 1 fX A/m 1/m

^ I * " I 5 1 J \ f ^ dt 5 t"*1) J - 2m Iz x o o

In other words, the integral

f 1 ( — |X |/(*) converges0 X 0

if and only if the series

m-1 k=m

where

v*

l <■ I % ) P converges

1/kt. = f |/(t)|^t . * 1/tf+i)

We say that t = belongs to a reverse Cesaro sequence space or

t ( d p .

Combining the above, we obtain that f € CES' where 1 < p < • if

and only if s = {s^> € c e s and t = {t € d’ where

rk rl/k8j. = J \f(t)\dt and t. = J |/(t) |dt .* V l * l/(fc+l)

Since the a-dual of ces is known and that of d can be found inP P

a similar way [23], therefore we have solved the problem of the Kothe dual

of CES .P

Theorem 7. The Kothe dual of CES for 1 < p < °° is the space of all

real-valued measurable functions g such that u = {uv } € ces a andk p

40

= es8-sup {| g(x)|; k-1 < x < k } ,

= ess-sup { |̂ (ar) |; (k+1)"1 < x < k"1 }

6. Some problems

We list in the following some problems which are being studied by my students and some Chinese mathematicians in Shanghai and Guangzhou.

Problem I. What is the Banaoh dual of X ?P

For an absolute sequence space, very often its Banach dual (i.e. the space of all continuous linear functionals) coincides with the a-dual, and also with the B-dual. But the space X^ is nonabsolute and its Banach dual includes the B-dual as a proper subspace.

Problem 2. What is the second 8-dual of X ?J PQ

We have characterised the B-dual X . The conditions on the d e ­ft P

ments in X^ are not symmetrical with respect to those in X . Hence thespace X is not perfect, i.e. X ^ t X . Then what is X 68 ?

P P P P

Some results on the above two problems have been obtained by Ou Zeming and Wu Bo-er.

Problem 3. Let X = {x; i4|x| € 7} and Y be solid. What is the a-dual

of X ?

Lim [12] has solved the problem for some special A and Y . The corresponding problem has also been solved completely for the nonabsolute case. It would be of interest to characterize the a-dual of X for more general A and Y . If Y is not solid, the problem seems to be harder.

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Problem 4. Let X be a Banaah space. Study the space ces^(X) of all X-valued sequences x such that

I ( “ I ll*J )P < + • »n=l n k=1

and similarly for the nonabsolute case.

The problem was proposed and is being studied by Wu Bo-er. For similar results on cQ(X), c(X), for 0 < p < - , and u^(^)for 0 < p < « , see [14] .

Problem 5. Let T be a family of infinite matrices and Y a sequence space. Let X be the space of all sequences x such that Tx € Y whenever T € T and

sup {|| Tx\\; T €1} < + - .

What is the &-dual of X ?

The above space X was studied by Lim and Lee [11] .

Problem 6. Let w d e n o t e the space of all sequences x such that there is a number I satisfying

1 n- I W - *| — * 0 as n ► » .” fc-1 K

What are the necessary and sufficient conditions on A such that an in­

finite matrix A maps into w■ for 1 < p < <» ?

This is a special case of a problem raised in Kuttner and Thorpe[8] . An attempt has been made in Lee and Lim [10] .

Problem 7. Characterize the continuous orthogonally additive function­

als on X .P

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Given a sequence space X , a functional / on X is said to be orthogonally additive if

f(x+y) = /(*) + f(y)

whenever x, y € X and x^ y^ = 0 for all k . Chew [l] has character­ized the continuous orthogonally additive functionals on I , Orlicz sequence spaces and a few others. It seems that the nonabsolute case is much harder than the following absolute case.

Problem 8. Characterize the continuous orthogonally additive functionals on ces

P

The above list, except Problem 4, coincides with that given at the end of a lecture series in Shanghai in 1983. The original Problem 4 was to find the a-dual of d' and was solved by Zhang [23] . The geometry of these spaces is also being studied by Yu Xintai and Zhang. For in­terest, we include in the references some standard reference books on the subject: [2], [3], [7], [18] and [26] .

REFERENCES

1. T.S. Chew and P.Y. Lee, Orthogonally additive functionals on normal lattices of sequences, to appear.

2. R.G. Cooke, Infinite matrices and sequence spaces, Macmillan, Lon­don 1950.

3. G.H. Hardy, Divergent series, Oxford University Press 1949.

4. A.A. Jagers, A note on Cesaro sequence spaces, Nieuw Arch voor Wis- kunde (3) 22(1974), 113-124.

5. A. Jakimovski and A. Livne, On matrix transformations between se­

quence spaces, J. Analyse Math. 25(1972), 345-370.

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6. P.K. Kamthan and Gupta Manjul, Sequence spaces and series, Marcel Dekker 1981.

7. K. Knopp, Theory and applications of infinite series, Blackie § Son 1928.

8. B. Kuttner and B. Thorpe, Matrix transformations of strongly summable series, J. London Math. Soc. (2) 11(1975), 195-200.

9. P.Y. Lee, Kothe duals and matrix transformations, SEA Bull. Math. Special issue 1979, 102-108.

10. P.Y. Lee and S.K. Lim, Some ideas involving strongly summable se­

quences, NUS Math. Dept. Research Report No. 128, October 1983.

11. S.K. Lim and P.Y. Lee, Matrix transformations between sequence spaces, to appear in Indian J. Math.

12. S.K. Lim, Kothe duals of some strongly summable sequence spaces,

I, II, and III , NUS Math. Dept. Research Report No. 137, December 1983; No. 141, February 1984; No. 146, March 1984.

13. I.J. Maddox, Elements of functional analysis, Cambridge University Press 1970.

14. I.J. Maddox, Infinite Matrices of operators, Springer-Verlag Lecture Notes No. 786, 1980.

15. P.N. Ng and P.Y. Lee, On the associate spaces of Cesaro sequence spaces, Nanta Math. 9(1976), 168-170.

16. P.N. Ng and P.Y. Lee, Orlicz sequence spaces of a nonabsolute type, Comm. Math. Univ. St. Pauli 26(1977), 209-213.

17. P.N. Ng and P.Y. Lee, Cesaro sequence spaces of a nonabsolute type, Comm. Math. 20(1978), 429-433.

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18. G.M. Petersen, Regular matrix transformations, McGraw Hill 1966.

19. W.H. Ruckle, Sequence spaces, Pitman 1981.

20. J.S. Shiue, On the Cesaro sequence spaces, Tamkang J. Math. 1(1970), 19-25.

21. J.S. Shiue, A note on Cesaro function spaces, Tamkang J. Math. 1(1970) , 91-95.

22. M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenrdumen, eine Ergebnisiibersicht, Math. Z. 154(1977), 1-16.

23. W.S. Polly, W. Zhang and P.Y. Lee, The dual of Cesaro function spaces, to appear.

24. A. Wilansky, Functional Analysis, Blaisdell 1964.

25. A. Wilansky, Swmability through functional analysis, North- Holland 1984.

26. K. Zeller and W. Beckmann, Theorie der Limitierungsverfahren, Springer-Verlag 1970.

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