An Or liez extension of Cesàro sequence spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seriii I: PRACE MATEMATYCZNE XXVIII (1988)

Lim Suat-Khoh and Lee Peng-Yee (Singapore)

An Or liez extension of Cesàro sequence spaces

1. In [1], Jagers determined the Banach dual of the sequence space bp for 1 < p < oo, where

bP = \x: % t \xk\}n>le l p},

k = 1

which is a slight generalization of the Cesàro sequence space cesp. In this paper, we generalize the lp space to the Orlicz space /ф, and also consider the so-called reverse Cesàro sequence space. More precisely, let \{1Ф)А\ denote the space of all sequences x such that A |x| e where A = (ank) is an infinite matrix given by a„k = <xk f}„, when 1 < к ^ n, and 0 otherwise. Similarly, |(/vy denotes the corresponding sequence space with  = (ü^) given by ank = akf}„

when n and 0 otherwise. We then determine the Kothe duals of and |(дл| subject to certain restrictions on the Orlicz function q> and on cck, fi„. We recall that the Kothe dual or а-dual of a sequence space X, denoted by X я, is the space of all sequences у — |yk} such that for every x = {xk| e X

00

X \xkyk\ converges.

k= i

We shall see that for Kg^l and |(/ч,)л| their Banach and Kdthe duals coincide.

2. Let (p be an Orlicz function (cf. [3]), i.e., q> is continuous and even on ( — oo, oo), convex on (0, oo), vanishes only at 0 and satisfies the following conditions:

(Oi) Ит<р(м)/м = 0,

u -*0

(°0i) lim (p(u)/u = oo.

u->00

We assume that (p satisfies the (A2, ^2) condition, namely, (p(2\u\) ^ M(p(\u\) for every и.

118 Lim S u a t - K h o h and Lee Peng-Yee

Let ф denote the complementary Orlicz function of q>, Le.,

\j/{v) = sup \\uv\ — (p(u); 0] .

We shall assume that there is a continuous increasing function / : [0, oo) -* [0, со) with inverse f ~ l such that the following three conditions are satisfied for u, u e[0, oo)

(2.1) /(и )и = <р(/{и)) + ф(и),

(2.2) v f ~ 1(v) = (p(v) + ij / (f ~i (v)),

(2.3) lim f ~ x(v) = oo.

V “ + 0 0

Note that if q>' is the derivative of (p, the function/is in fact (<p')~l . For example, when (p(u) = up/p and ij/(v) = vq/q, f ( u ) = uq~ 1.

Now consider the space \(lv)A\f where ak > 0 for every k, /„ > 0 for every n and \P„}„2i 6 lv . Using the Orlicz norm in ltp, with the usual notation ||-||°, the norm in K /jj is defined by

iwi = m w ii;.

We shall also consider the space

\ ( Ш = {*’• % £

k = n

where the norm is similarly defined by imi = m m C

It may easily be verified that with the norms given above, the spaces

\(lv)A\ and КУя! are BK spaces (cf. [2]). In fact, they are absolute in the sense that if хеК/ДЛ and \z\ = |x|, then геК/Д*| and ||z|| = ||x||. This means that the Kothe dual of | ( y j coincides with its /-dual (cf. [2]), and similarly for К Ш - Also, for y e | ( y x

(2.4) £ K y J < 11*111Ы1*.

k = 1

where

Ы \ р = sup {| £ xk yk| ; xe|(/„)J and ||x|| < 1}

* = 1

s u p { £ k h l ; *е|(/Д<1 and Цх|| < 1}.

k = 1

This is also true for |( у л| and its Kothe dual.

As for notation, we shall adopt the usual notation used for Orlicz spaces such as

00 00

M s) = X И Ы ) and 0*(f)= X

f c = l k = l

and shall use without proof known identities such as

INI? ^ M s)+ i-

3. Now, we shall characterize the Kothe dual of \(lv)Al First, for any sequence у — {yk} satisfying y„]n>1 e c 0, define y~ to be the coordinate- wise infimum of all y* ^ |y| such that

(3.1) {an 1y*)n^i is decreasing

and 'f t C /C O I ^ i is increasing, where

t* = ft"1 («п 1 У* ~ <*й+1 yï+i).

Next, let С = Р й 1(!Хй 1УЙ— ЯЙ+1УЙ+1) and define the sequence

\m(k )}k>1 by

m( 1) = min [n; j8n~ 1 /(r ~ ) > 0},

m(/c) = min jn > m ( k - 1); р й 1 / ( C ) > Рй- i / ( C - 1)} •

Let / k = m(k)+ 1, ..., m(k + 1) — 1] and then, for each к, there is a constant ck such that

(3.2) ^И_1/ ( С ) = for every n e Ik.

Note that ck increases strictly with к and, for n = m(k) + 1, ..., m(k+ 1), (3-3) аЙ1УЙ = C (i) Ут(к) - X ft / '"1 fat ft)

j = m ( k )

and, in particular,

(3-4) am(^ y m(k) — otmik +1) ym(k +1) = %к ft / 1 if к ft)»

where Гк denotes the sum over all j in Ik. If the sequence 'm(k)} terminates at m(K), equation (3.3) holds, with к replaced by K, for all n > m(K).

3.1. Le m m a. For every к such that m(k) is defined., y~(k) = |ym(k)|.

Proof. Suppose that for some к ^ 1, y~(k) > |ym(k)|. By continuity and monotonicity of f since £~(jS)/(C(k)) > f t ^ - i/X C w - i) » there exists ^e(0, *) such that

f (Pmljc) (a m(fc) (/Vm(fc)) — am(k) + 1 Ут(к) + 1 )))

> f t n ( k ) - 1 J ( f t n ( k ) - 1 ( a m (k )- 1 Ут(к)~ 1 — a m(k) ( / У т ( к ) ) ) ) > 0

120 Lim S ua t - Kh o h and Lee Peng-Yee

and we may ensure that Лу~(к) > |yw(fc)|. Defining y* by y f = y~ for j Ф m(k) and y*(k) = /у„Т(к), У* satisfies (3.1) but y*(k) < y~(k), which gives a contradic­

tion.

3.2. Le m m a. For any sequence у satisfying {oc~1 y„}„^1e c 0, a ~ 1 у ~ —> 0 as n —> oo.

P roof. Since ja"1 Ул~}л2м is non-negative and decreasing, it is a con­

vergent sequence. If the sequence \m(k)\ does not terminate, a “(k)y~(k)

— am(k) |ym(k)l 0 as к ->co. Hence \(x~l y~}n>1 is a null sequence. Now we consider the case where (m(k)} terminates at m(K). Assume that a ^ y ^

-> c > 0 and let M > m(k) be a positive integer such that a ~ 1 |y„| < c/2 for n > M.

Let g be a function defined on the reals by

9 ( v ) = х м 1 Ум~

Z ft/'^ft)-

j = M

Since is increasing in v for each j, the series is uniformly convergent for v in any finite interval and hence g is continuous in v over any finite interval. Replacing к by К in equation (3.3) and taking limits as n -> oo, we have

c = lim ct~ 1 y~ =g{cK) n-*oo

and for v sufficiently large, g{v) < 0. Hence there is a positive number L > ck such that g(L) = c/2. Defining y* by у* = Уп for n ^ Af and

a*'1 У* = «м1 У м -

Z ft

j — M

we have a sequence y* with y* < y ~ for n > M but y* satisfying (3.1) which is a contradiction.

3.3 Lemma. I f (a"1 y„}n>i e c 0 and t ~ e l ф, and if s = A |x|g/^, then ft" 1 a„~1 Уп -+ 0 as n —> oo.

P roof. By Lemma 3.2, we may write

00 00

ft'1 <*„1

ft"1

X («*'1

~ <*ГЛ

iK Z 5*

k = n k = n

Since s = A |x| el<p, and t~ е1ф, the last series above converges to 0 and the result follows.

3.4. Th e o r e m. Let ye\(lv)A\* with ||у||д < 1. Then (i) a “ 1 y„ —► 0 as n -* oo and

(ii) Г = у ~ Л ~ 1е1ф.

Proof. First, define .y(/7) = a ' 1 e", where en is the sequence with 1 in the nth position and 0 elsewhere. Let 0 = \fik\k> \ and 0n be its truncation at the nth term. Then A |x(n)| = в — вп~ 1 and hence ||x(n)j| = \\0 — 0и-1||°. Since is an A K space (cf. [2]), ||0 —0"~ 0 as n -» oc and, substituting x = л:(п) into equation (2.4), condition (i) follows.

To prove (ii), first consider the case where m(k) terminates at m(N).

Writing c0 = 0, define the sequence y by

Then

X m(k) i^k l)

xt = 0,

к = 1, 2, 3, ..., N, otherwise.

if M n < N, if M„ ^ N, where M„ = max \k\ m(k) ^ n\.

Since Al xlel y and so x e |( / ^ |. For n e l k, M n = k and so, for к = 1, 2, ..., JV,

k c Mn = P„ck = / ( C ) and, for n ^ m(N),

P„Cn = / ( 0 - Hence A\x\ = !/(?„"“) and thus

(3.5) M I /( C ) ! n » i) = < x -

Now, substituting к = N in (3.3) and letting n-^oc, by Lemma 3.2, (3.6) ат(Н)Ут((\) = Z P j f 1(CN^j)'

j = m ( N )

Then, using Lemma 3.1 and equations (3.4) and (3.6),

(3.7) |И |х |||;г Н |х |||Ы |'» £ I x ^ i r t , ,

k = 1

= I ^ . W i a j + c , £ / v ~ w , >

k = 1 j = m( N)

m(N) — 1 oo

= Z М / ( * Л ) + М Л ] + Z

j = 1 j =rn( N)

the last equality resulting from equations (2.2) and (3.2).

122 Lim S u a t - K h o h and Lee Peng-Yee

Since i) is finite, and \\A \x\ ||J < Q ^ f {tf)\j ^ 1) + 1, M O 00

= ]Г <A(0 < 1 which gives (ii).

j = i

Now, if the sequence \m(k)] does not terminate, let y(N) be the sequence у truncated at m(N) and let r(N)~ be the corresponding sequence. Then for y(N), the \m(k)) sequence terminates at m(N) and yfc~

= y(N)k for к ^ m(N). Hence for j ^ m(N), t f — t ( N ) f and from above, m(N)-l m(N) — 1

I ф ю = z Ф ( т г ) < l.

j= i j= i

Since this is true for every positive integer N, we obtain M O ^ 1 by letting N -» oc and so t ~ e 1Ф.

3.5. Th e o r e m. ye\(l9)A\* if and only if (i) а ~ ху„- ^0 as n -*■ oc and (ii) f~ е1ф, where t ' — (/.у)" A ~ x for some X > 0.

Proof. First, we show the sufficiency of (i) and (ii). From (i), а ~ 1(Яу„)~

-» 0 as n —* oo, by Lemma 3.2. Hence, writing s = A |x| e and applying Lemma 3.3 with Xy replacing у to the following:

n n It— 1

Z \хк*Ук\ < Z i**K4y)r = Z sk*k + ^ ‘ * ; 1( ^ ) ; * л;

k = 1 k = 1 к — 1

we have Xyel(lv)Aja. Hence уеК/ДЛ*.

Now, conversely, if ye\(l9)AГ take Я = 1 if ||yj|^ ^ 1 and X = (IMP)-1 if IMP > 1. Then conditions (i) and (ii) follow from Lemma 4.1.

4. Now we shall characterize the Kothe dual of |(/^)д|

CO

= {*: \P* Z ak\xk\}n^i^i<p}i where ак > 0 for every k, pn > 0 for every n.

k = n

We further assume that for any constant scalar у > 0, n

(4.1) Z ф ( / “ 1W ) -* 00 as n —* сю.

j= i

4.1. Le m m a. I f y e \{19)a\\ г^еп l^ere Is a positive number В such that, for every n,

(4.2) А Г 1ТО)1МР-

J= 1

P ro o f. First, choose a positive number В such that ф ^ ~ х ( B ^ ) ) ^ 1.

Then, for every positive integer n,

(4.3) Z « /'( /- 1W ) ^ 1.

j=i

Defining the sequence x(n) = (a„ ]T P j f ~ l {BPj))~i en, it may be checked that i= i

Л|х(м)|б/<л with

11х(«)ц = |И |х (П) | | | ^ [ £ <pW j)+ £ ^ ( / - ‘ в д Я ' М ! <pW j) + i)

j= 1 j= 1 i= i

from which ||х(и)|| < 1 by (4.3). The result follows on substituting x(n) into (2.4) .

Now, for any sequence y, let y* ^ |y| satisfy

(4.4) { a i s increasing and [fi^1 f (u*)}n>1 is decreasing, where

(4.5) u* = f a 1 (a ~ 1 y* - a "Л у*- r) and a ô 1 y$ = 0.

For any sequence y satisfying the following condition:

There exists a positive number В such that (4.6) . \yn\ < a„ f p j f - ' i B p j ) for all n.

j= i

We may define y* to be the right-hand quantity of inequality (4.6). Such a y*

satisfies (4.4), and hence such y* exist for sequences у satisfying (4.6). Let y~

be the coordinatewise infimum of all such y* and define u~~ for y" as in (4.5) with a ô ^ o = 0 . The sequence \m(k)\ is defined by

m(0) = 0, m(k) = min {n > m ( k - 1); f ( u ~ ) > 0 ~ + i/(m~+ i)}.

Now, for each к such that m{k) is defined, there exists a constant ck such that (4.7) 0 ~ 1 / (u~) = ck for m (к- 1 ) < n ^ m (к),

where ck decreases strictly with k. Hence, for m(k — l ) < n ^ m ( k ) , ' П

ап_1уГ = a ~ (i _ 1)y~(k_ 1)-f X f i j f ~ 1(<:kfij)

j ~ m(k — 1 ) + 1

and in particular,

(^•^) ^m(k) Ут(к) ^m(fe- 1) Ут(к- 1) = f i j f ((-к fij) >

where t k denotes summation from j = m(k - 1)+1 to j = m(k).

Furthermore, if the sequence {m(k)} terminates at m(K), then for every n > m( K),

(4.9) f i n 1 / ( * 0 = ^ + 1

and

П

(4.10) <X~ 1 у,Г = У.~(К) Ут{К) + Z f t / '" 1 (CK + 1 ft)•

j= m(K) + 1

4.2. Le m m a. //у е К У д Г w /f/i ||y||^ ^ 1, then for any positive scalar y, Г»"1 W < Z Ф(vft) + 1 far all n.

y= i

P roof. The result follows by substituting x(n) = ya"1 en into (2.4) since

A |.x(«)| = у 'ft, f t , ..., ft, 0, 0, ...) and

124 Lim S u a t - K h o h and Lee Peng-Yee

||.y(/7)|| = 11A |.y(w)| ||J < ^ ( Л |х ( л ) |) + 1.

4.3. Le m m a. Let ye\(l,p)Ala and ||y||^ < 1. I f the sequence \m(k)\k>l ter- minâtes at m (K), then for n > m(K)

У(п) = Ck 1 1 Ун Z Ф(ск

j= i +1_{f t )}

increases with n and lim g (n) exists.

n-*rf

Proof. Since g(n) — 0 if cK + i = 0 and the result is trivial, it will be assumed that cK+, > 0. First, note that for any n > m(K), by (4.9) and (4.10),

Z Ф(иП = Z <А(/ 1 (Çk + 1 ft))

m(X) + 1 m(K)+l

= Z CK + l P j f 1 (CK+1 ft)~ Z ^ (CK+lft)

m(K) + 1 m(K) + 1

n

= C K + 1 ( а и 1 Уп — a m(K) У т (К )) — Z Ф + 1 f t )

m(K)+ 1

n m(K)

= c'jf f 1 a n 1 ~ Z ^ + 1 f t ) + Z Ф + 1 f t ) — CK + 1 am(K) Уж(*) •

Hence (4.11)

m ( K )

z1

Z Ф(и^ ) —в(П)А- Z ^ + 1 ft) “ + 1 ат(К) Уги(К) m(K) + 1

and so g{n) increases with n > m(K). Now,

(4.12) g (n) = cK + ! ot„ 1 \ynI - X <P(cK + i ft) + ^ + 1a / (у,Г - |y„|) j= i

^ l + c K + i

by Lemma 4.2. Now consider two cases:

C ase 1. There is a subsequence \y^k)\k^i such that = |y„(k)|. Then, for all к such that n(k) > m(K), g ( n ( k ) ) ^ 1 by (4.12), and since g(n{k)) increases with k, lim g(n(k)) exists. But, as \g(n))„>x is an increasing

k~* oo sequence, lim g (n) exists.

C ase 2. There is a positive integer M ^ m(K) such that yÙ — \yM\ and Уп > \Уп\ for all n > M.

It will be shown that either a y* may be constructed to obtain a contradiction or that cK+l = 0 which is again a contradiction. Now, for each

П

n > M, a^ 1|yJVf|+

X

M + 1

an l \y'n\ > <*м1\Ум\, define c(n) to be the positive number such that

(4.13) an"1|y„| = *м1\Ум\+

X

M + 1

If a„_1 \У„\ < «м1 ITmI. define c (n) = °- Now, since

a n 1У a = «м1 Ы + X f t / 1 (ск +1 ft), м +1

we have 0 ^ c{n) < c K + 1 for all n > M.

Now define c = sup [c(n); M).

C ase (i). If c < c K + 1, define y* by y * = for n ^ M and

« ; ‘ Й = *мЧум1+

i

and obtain a contradiction as y* satisfies (4.4) but |y„| ^ y* < y~ for n > M.

C ase (ii). If c = cK + l , since c(ri) < cK + l for all n > M, we must have a countable subset S x of positive integers such that lc(n))neSl is convergent to c. We will show that

c = 0 and hence cK + x = c = 0.

If c(n) = 0 for all large n e S x, there is nothing to prove. Hence it may be

126 Lim S u a t - K h o h and Lee Peng-Yee

assumed that there is a countable subset S2 of S x such that c(n) = 0 for n e S l \ S 2 and (4.13) holds for n e S 2.

Then, for every n e S 2, using the proof of Lemma 4.2 with y = 1,

«м1 \Ум\ + Z f t / ~ 1 И ") ft) = a«~1 W < \\ {Pj}i *j*X

M + l

which implies

Z c(")ft/- 1 (c(")ft) < llMn)ft}i^Jj < Z Wc(")ft )+1

M + l j = 1

and so

Z <?(c(")ft) + Z H /^H ^ft))^ X P(c(”)ft )+1

M + l M + 1 j' = 1

giving

(4.14) £ </'(f~, (c(n)0j)) « 1 + 1 4>(c(«)ft)

M + 1 j = 1

M

< i + Z <*>(<*+1 ft) = я (say)-

;=i

Now, if c(n) does not tend to 0 as n -> oo in S2, there is а у > 0 and a countable subset S3 of S 2 such that c{n) > у for all n e S 3. Hence for n e S 3,

Z <A(/~ 1 H w)ft))^ Z *А(/_1№)-

M + l M + l

But the right-hand side is divergent as n -*■ oo which contradicts (4.14). Hence {c (rc))„6s2 is null sequence. But c(n) = 0 for every n e S t \ S 2. Hence {c(n)}neSl is also a null sequence and so cK+1 = c = 0 which gives a contradiction for case (ii).

Thus Cases 2 (i) and (ii) are impossible. Hence the result holds, i.e., lim g(n) exists.

4.4. Th e o r e m. Let with ||y |p < 1. Then у satisfies П

(i) there exists В > 0 such that |y„| ^ a„ Z f t / - 1 (£ft) f or n * (ii) y ' A 1 = и ' е1ф. j~ i

Proof. The first condition follows from Lemma 4.1. With (i), y ' is well defined. Consider first the case where {m(k)} terminates, say at m(K). By equation (4.11) in Lemma 4.3,

n m(K)

lim Z ф{ и ? ) = { й т д { п ) ) + Z <P(ck + i f t ) - c * + i «ык) У *ю-

П-+ oo m(K) + 1 n (X) 1

Therefore, by the result of that lemma ^ ( u f ) converges and hence m(K) + 1

и ' е/ф.

Now consider the case where \m(k)} does not terminate. Let N be a positive integer and define x as follows

Хт(к)=*т(кЛСк ' С к+1) for к — 1, 2, N 1, x m(N) — a m(N)

Xj = 0 otherwise.

Let Mj — min [k; m(k) ^ j}. Then

14 f PjCM1 if M j ^ N ’ ie > j ^ W(N)’

X J ( 0 otherwise.

Hence |x|e|(/vy and

N N m(N)

q<?(a M )= Z Z ^ ( / ( « Л Н Z <Л/(мЛ)-

k= 1 k=l j= l

m(ff)

Thus, IMI I I # ^ \\A |x| HJ < Z <p(/K ~ ) ) + 1 and furthermore, y=1

oc N

Z l^k-Vkl = Z (am<k) >m<k) ~ am(k-l).Vm(k-l))Ck k = 1 fc = 1

= Z 2 ы Ш ~ Ч ъ Ь ) ) с к fc= 1

= Z 1 к(я>(скР]) + Ф ( / ~ 1 (Ск^)))

k= 1

m(N) m{N)

= Z <р( /( « Л ) + Z <ИмЛ -

7= 1 J= 1

m(ff)

Hence, substituting into (2.4), Z Ф(иЛ ^ L Letting N -* oo, дф(и~) ^ 1 and

~ / j= 1

so u е1ф.

4.5. Theorem. уе|(/(/))д|“ if an d only if for some Я > 0,

Л

(i) t/iere exists В > 0 such that À\yn\ ^ a„ Z! f or aH n '->

(ii) (ХуГ{Л)~1е1ф.

Proof. If conditions (i) and (ii) hold for some Я > 0, since

Z М 'Ь 'к Ж z « Г ( ^1^1)к-^»Г1а ~ 1(Яу)~(Л|х|)п ^ \\A |x|||J||ir||,,

k= 1 к = 1

128 Lim S ua t - Kh o h and Lee Peng-Yee

where u = { À y ) ' A ~ \ ly e \(lv)A\* and so y e |(Ц*Г. Conversely, if y e |(/„ЬГ, choose л — 1 if ЦуЦ** ^ 1 and л = (||y||0_1 if 1Ы1Д ^ 1- Then (i) follows from Lemma 4.1 and (ii) from Lemma 4.4.

5. The results in Sections 3 and 4 charaterize the Kdthe duals of K/^J and I(l,p)A\. Of greater interest may be the special cases in which A is the Cesàro matrix C, where ctk = 1 and fin = l/n and when À is the so-called reverse Cesàro matrix whence ak = 1 and /?„ = n.

References

[1] A. A. J a g e r s, A note on Cesàro sequence spaces, Nieuw Arch. Voor Wiskunde, III Ser. 22 (1974), 113-124.

[2] P. K. K a m th a n and M. G u p ta , Sequence Spaces and Series (Marcel Dekker), 1981.

[3] M. A. K r a s n o s e l s k i l and Ya. B. R utickiT , Convex Functions and Orlicz Spaces (Noord- hoff, English Edition), 1961.

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