oo Superposition operators on w and

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ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIX (1990)

Chew Tu a n Seng (Singapore)

Abstract. Let w0 be the space of all sequences which are Cesâro strongly summable to zero.

Characterization of superposition operators acting from w0 to /j is given in this note. The similar result for the function case W0 is also given.

1. Introduction. Let R be the set of all real numbers. Let Z + be the set of all natural numbers. Let S be the set of all real sequences. Let g(k, t): Z + x R ->• R such that for each k e Z +, (1) the function g(k, •) is continuous and (2) g{k, 0) = 0. Let

The operator Pg is called a superposition operator. Characterization of Pg on Orlicz sequence spaces was given by J. Robert and I. V. Shragin [6], [7]. In this note, we shall characterize Pg on w0 and state without proof the similar result for the function case W0. Our proof also works for Orlicz sequence spaces though it differs from that of Robert and Shragin [6], [7].

Their proof is elegant but it does not work for the space w0 due to the difficulty in providing an argument by reductio ad absurdum. However our proof using the continuity of orthogonally additive functionals may be lengthy but it applies to both Orlicz spaces and w0 and it is constructive in nature.

2. On w0. Let w0 be the space of all sequences which are Cesàro strongly summable to zero, i.e.,

Superposition operators on w 0 and W0

00

/j = {xeS; £ |xk| < oo with x = {xk}}.

k=i

Define an operator Pg from S into S as follows:

Pe(x) = {g(k, xk)} for every xeS .

3 — Roczniki PTM — Prace Matematyczne XXIX

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150 C h e w T u a n S e n g

It is well known [4], p. 170, [5], p. 20, that w0 is a Banach space with norm

IIII* = sup- £ \xk\.*

n П к = 1

Furthermore, this norm is equivalent to the norm Ml = sup {2~rX rkl}

Г

where denotes a sum over the range 2r ^ к < 2r+1 and r = 0, 1 ,2 ,... In this section, we always use || • ||.

Th eo rem 1.

An operator Pg acts from

^{w 0}

to iff there exist

{c k} e

and { d je /j with ck ^ 0, dk ^ 0, and rj > 0 such that, for 2r < к < 2r+1, r = 0, 1, 2, ..., we have

\g{k, O K dk + cr2 r |l* whenever |t| ^ 2r rj.

The proof of the sufficiency is easy. To prove the necessity, we need the following two results.

L emma 1. Let g(k, t): Z + x R ->■ R satisfying (1) and (2). I f there exist a > 0

and P > 0 such that for any finite sequence x = {xk},

1 И * . x k)I < « к

whenever |xk| ^ ft, then there exists { cj

e

f with ck ^ a and ^ 0 such that for each k,

\g(k, 01 ^ ck + 2aP- 1 whenever |t| < P/2.

P ro o f. The proof is similar to the function version [3], Lemma 17.6. For each k, we define

h(k, t) \g(k, t ^ o i p - ^ t ] if \g(k, t)\>2<xp~l \t\,

0 otherwise.

For each k, let

ck = sup {h(k, t); \t\ ^ p/2] = h(k, zk).

Note that the function h(k, •), which is continuous, assume its supremum. For each m e Z +, we can decompose the following sum

m

X Iz k\ == ^ 1 + T 2 + . . . + Г , k = 1

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so that

0/2 for i = 1,2, . . . , / - l , and 0 ^ ^ 0. It follows that

m m m

E

c k =

Z zfc) |- 2 a 0 -1

X

|zk|

k = 1 к = 1 к = 1

^ lot — 2a/?-1 02- 1 (/ — 1) = a for every m e Z +.

Hence { c je /j with Ekck ^ a- It is clear that, for each к, if |£| < 0/2 then h(k, t) ^ cfc. Note that

h{k, t) = Ig(k, t)\ — 2(x[$~l \t\

if Ig(k, 01 ^ 2a0-1 \t\. Therefore

\ g { k ,t) \^ c k + 2ap~1\t\

if |t| ^ 0/2. It is clear that the above inequality still holds if \g(k, 01 < 2a/?- 1 \t\.

Hence, the proof is complete. ■

R em ark. The way we decomposed m

I |2Tj = 27^2 7 ,+ ...

k= 1

in the above Lemma is different from that we did for the function version [3], Lemma 17.6. Recall that, for the function version, given any bounded measurable function f let

* b

§\f(x)\dx = np + G

a

where 0 < e ^ 0 and 0 is given. Then we can divide [a, b] into n + 1 subset, say X lt X 2, ..., X n + 1 such that

{ |/(x)|<frc < 0 for i = 1

^,

2 ,..., n + 1

^.

However, this cannot be done for the sequence version.

A functional F defined on w0 is said to be orthogonally additive if F{x + y) = F(x) + F(y)

whenever xk yk = 0 for each k, where x = {xfc} and y = {yk}. The following representation theorem is given in [1], Remark 2.

Th e o r e m

2. A functional F defined on w0 is orthogonally additive and

norm-continuous iff there exists g ( k ,t ): Z + x R - + R satisfying (1) and (2) in

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152 C h e w T u a n S e n g

Section 1, and (3) Pg: w0 -> f such that 00

F(x) = £ g{k, xk) for all

xgw

0.

k = 1

P r o o f o f th e n e c e s s ity of T h e o re m 1. If Pg maps w0 into f , then, by Theorem 2, the function F defined on w0 defined by

F (x) = X |0 (fc, xk)| for all x = {xk}

gw

0

к

is orthogonally additive and norm-continuous. Hence, there exists rj > 0 such that, whenever ||x|| ^ rj,

2 > ( k , **)l ^ 1- к

For each r, there exist ук, к = 2r, 2r-h 1, ..., 2r + 1 — 1, such that Ук)I = suP{Zrl0(fe> Ч)\’ 2_гХг1У < »/}•

Let ak = \g(k, yk)|, к = 1, 2, ..., then we claim that Xk°=iflk < 1. Indeed, for any m — 2r, r = 0 , 1 , 2 , . . . , we have

m

X Tk)l ^ 1 for a11 w

к = 1

since the sequence z = {zk} with zk = yk, for к = 1, 2 ,..., m and 0 elsewhere belongs to w0 and ||z|| ^ rj. Consequently, Xfc°=iak ^ 1» i-e-> {ak \e h-

On the other hand, for each r,

Xr tk)\ ^Z r« k

whenever 2~rYjr\tk\ ^ 1- Apply Lemma 1 to the above inequality with* a = Z rak> P = 2rl ап^ 9(k> t) — 0 for к Ф 2r, 2r+ l , ..., 2r+1 — 1. Note that if* X k k l ^ A then

Z r W ^ Х к К £ . к

Hence

Zri(. xk)l = Х И ^ xk)i < a- к

So, there exist bk ^ 0, к = 2r, ..., 2r+1 — 1 with Xr &k ^ Zr ak — a such that, for 2r ^ k < 2 r+1,

\g{k, t)I ^ bk + 2 ( ^ rak)2~rrj-1 |t|

whenever |t| ^ 2r r\/2. The proof is complete. ■

3. On W0. The similar result for the function case W0 will be stated in this

section. In view of its similarity to the sequence space w0, we shall not give the

proof here.

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Let

T

W0 = |/(x)|<*x = o l .

Let k ( x , t ): [1, oo) x R - + R be Carathéodory function, i.e., the function k(x, ■) is continuous, for almost all x e [ l , oo) and k(-, t) is measurable for every te R . Furthermore, let k(x, 0) = 0 for almost all x e [ l , oo). Define

T

heorem

3. An operator Pk: PF0 ->L1( 1, oo) iff there exist a > 0 and {cr}

e

f such that for each r, there exists

g r

e L t [A (r)) with | Д(г)

\gr

(x)| dx ^ cr such that for almost all xeA (r) and all t,

[1] C h ew T u a n S en g , Characterization o f orthogonally additive operators on sequence space, SEA Bull. Math. 11 (1987), 39-44.

[2] A. К a m in ska, On comparison o f Orlicz spaces and Orlicz classes, Functiones et Ap

proximatif) Commentarii Mathematici 11 (1981), 113-125.

[3] M. A. K r a s n o s e l ’sk ii, Integral operators in spaces o f summable functions (translation), Noordhoff 1976.

[4] I. J. M a d d o x , Elements o f Functional Analysis, Cambridge University Press, Cambridge

[5] L ee P e n g Y ee, Sectionally modulared spaces and strong summability, Commentationes Math, Tomus specialis in honorem Orlicz, vol 1, 197-203.

[6] J. R o b e r t, Continuité d ’un opérateur nonlinéaire sur certains espaces de suites, C. R. Acad. Sci.

Paris, Sér. A. 259 (1964), 1287-1290.

[7] I. V. S h r a g in , Conditions for imbedding of classes o f sequences and their consequences, Matem. Zametki 20, 5 (1976) (in Russian), 942-948 (English translation).

Pk( f № = k(x,f(x)).

\ k( x, t ) \ ^gr(t) + ot2 rcr\t\

where A (r) = [2r i, 2r).

References

1970.

DEPARTMENT O F MATHEMATICS, FACULTY O F SCIENCES, NATIONAL UNIVERSITY OF SINGAPORE, SINGAPORE