Sublinear functionals and Knopp's core theorem

(1)

Internat. J. Math. & Math. Scl. VOL. 13 NO. 3 (1990) 461-468

SUBLINEAR FUNCTIONALS AND

KNOPP’S CORE THEOREM

C.ORHAN

l)ei’;utmcntofMathcrmtics, l:ac’.ItyofScic.c," UniversityofAnkara, Ankara-()61(10, "I’U

R

K

F.’f

(Rece’ived January 26, 1989 and :l.n revised form October 20, 1989)

A

BSTRA(YI’.

In

thispaperwearc concerned withinequalities involvingcertain sublincar functionals on m, the spaceofreal bounded

sequences.

Such inequalities being analoguesof

Knopp’s

(.’trc theorem.

KI:.Y Wf)RI)S ANI)iliRASFS. (;oretheorem,Sublinear function:lls, inl’i,ilc ,,:trice,,, convergence.

1980

AMS

Ci.,ASSIFFICATION

CODES.

40C05,40J()5,46A45

!.

INTRODUCTION.

l.clmbe timlinearspaceof realbounded

sequences

withtheusu:istq)rcmumnom.Wewrite n

m

0={xe

m’suplE

Xkl<Oo}

n _k=0

Let 1.

be thesequenceof infinite matrices(A

i)

(ank(i)).

Given a

sequence

x (x

k)

wewrite

A(x)=

Z

ank(i)

x_k

k=O

ifitcxistsforeachnandi>0.

WealsowriteAxfor(Ai

)oo

Thesequcnccx=(xk)

i’ said tobesummable

m

the value sbythemethod

(1.)

if

n(X)

i,n=O"

Atn(X)

s (n ,,*,uniformlyini) (1.2)

If (1.2) holds, thenwe write

xs(,.).

If we define

(ank(i)

by

ank(O

_otherwise<k

<

i+n

then

(1)

reducestothe mctiodf (Lorcntz 11).

In

the case i+n

ank(i)

E.

ark

r=l

(2)

462 C. ORHAN

Themelhod

(1.)

is saidtobe :onservative l’x impliex s’().If )i,c,n’,eraivc and s’,lhen

()

iscalled regular.

It

iswell known, (Sticglitz 131), that

()

isregularifan(!only if thetllxing hold:

X

ank(i)1<

(fi,r all n, for alli). k

andthere existanintegermsuch that sup

lira

ank(i)

0, uniformlyin i, I.’1

n

liraT.

ank(i)

unifinnlyini. (1

n k "i’hrough(,ttile

paper

we write

I11

sup

k

ank(i)

<

n,!

tomeanthat,thereexists aconstant

M

suchthat

Z]

a,lk(i)

<M

k and tile series

(for alln,for all i) I.,

E

ank(i)

k

convreges unifonrflyin for each n.

If, foreveryboundedsequencex,

x---s(l.)

then

(,,,’1.)

issaidtobe a Schur Throughoutthepaperwe consideronlyreal matrices and real tmndcdscqucncc,.

Inthispaperwearcconcerned withinequalities involvingcertain sublinearfnctimal,;m

.

thespaceofreal boundedsequences.Suchinequalities being analoguesof

Knopp’s

(’orehcretn Th:ttheorem determines a class ofregularmatrices for which

limsup

Ax <

limsup

x

for all xe m, seee.g Cooke

I41,

Maddox

151,

Simons

161.

Thisresult has also beencxtcntlcd corcgular matricesbyRhoades171,Schaefer181,and,Das

191.

Beforestatingtiletheoremstobeproved,we introduce some flrther notalion.

[(x)

_{liminfx n} L(x) limsupx_n Ilxll sup

Ixnl

i+n

[.*(x)

limninf

su.p

n

+ _r=i

E

x r i+n L*(x)

limsup

su.p

:

_{x r}

n _r=i

W*(x)=

i:f L*(x +z) z m₀

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SUBLINEAR FUNCTIONALS AND KNOPP’S CORE THEOREM 463 If f,garc

any

twooftheatx)ve_functionals,_we_{shall write}_fA<_

gB

todenote that, forevery I))tll(lctl .,,tlUClt’x, tletran.,l’orms

Ax

and

Bx

aredefinedan(Iboltndcl aml f(Ax)< g(llx).

2.’1’1 I1"rl,,\lN RI!.’qUI.’I’S. Wt"

,,.

itc, Itxti Ill,

QA(x)

limsupn

st’tl>

ank(i)

xk

dil(l

qA(x)

liminfn

st!p

ank(i)

x k \Vltt ttlt.,,totation wehave

"1"!ll..’()RI:.M.I.

Let

I11.11<,:,o. Then

1 aid,lyif(,,A.)isregularand

QA

<L

_(2.1)

T[

a,,k(i)

--

(n--),tmifomlyi,,i)

(2.2)

k

I’R()()i:. Necessity. l.ct x (x

k)

beac()nvcrgcnt .,,etltcnce."rhcn[(x)--l.(x)= libel x. l}y 2. ), wcIavc

[,(x)

<

QA(-X)

<

QA(X)

< L(x). Icltcwe getthat

QA(X)

tlA(X

lirax.

So (,A.)

isregular.

.’,;i,cc(..A.) isregular,therequirementof i..emma2, (Das191), is satisfied,ltence there e\i.,,t.,,_ye

.

such thatIlyll < and

QA(Y)

linsui) sup

1

ank

{i)

I-lic,i.c,taktlg x c (1,1 ),welmve

--qA(C)*_"litninfSUl)

T.I

a,k(i)

kl

<

limsup sup

k’Xl

ank(i)

QA(Y)

<

L(y)<

iiyil < n

w,_llci_i)r)vc.,,_{the necessity of}_(2.2)

.";tllticicwy.

Wc

dcl’inc,for;ilay real

Z.,

*

max

(X,()),

X."

i.ax(-*,1))."l’il.

I;kl

:k+

-

*

;iiltlA )L

L’.

lcncc

T_.,a,k(i)

xk=

E

ank(i)

x

k+ E

ank0)

_{x k-}

E

a’lk0)

_{x k}

k k<m k>m k>m

Y’.a,,k(i)

x k<llxll

Z

hnk(i)l+

(sup

Xk)

k k<m k>_m k>_m

By

hypothesis,we get that

QA(X)

< L(x).

(4)

464 C. ORHAN Cf)R()IJ,ARY.2.Wehave on m,

[<I,* <1,*<i,. PR()()F.inIheorem 1.ilisenoughtolake

l/n+

ank(i)

₀ _otherwise_<k _<. i-n

We

de(luteatonce from

Corollary

2that if a

sequence

x isconvergenltos. tllelv itisaltvu)’,t convergenttoswhich is awell-knownresult.

Wenotethat,by consideringl’heoren) l, onemaygetnecc,,aryanti

,,

fficicnt(’m(liti,r,I,,v

,*A_<!,andI.A.<_

inthenext Iheoren)weconsidertheinequality I.A <1.*.

TI

IEOREM.3.

LA

_<I,*if and

only

if

A

isstrongly

regular

and

E

bnk

--

(n-+ ,:,,,) (2 )

k

PROOF.

Recall thatamatrixA is calledstrongly regularif it mapsall se(luencesintothe convergentsequences andlira

Ax

f- lira x.

Wefirstprovethe necessity, itiseasytosee that[.* <[,A<I.A<!.

.

If x i,alvu,,,t c(,ivergev!

then f- lira x

[.*(x)

L*(x). tlence, bythe hypothesis,

[.(Ax)

L(Ax) f- lira x. SoA

.,

strongly regular. Using thefact thatL*

<

I,(see Corollary 2)and that

LA

< I.,*, we get float

IA

5I..

Now

tilenecessity of

(2.3)

follows from

Knopp’s

Core

theorem(see,e.g.IVlatldox151).

We

noteinpassingthal a matrix

A

is

strongly

regular,

Lorcntz

[l ].ifan(l_(relyif ii,vcgtlar andthat

Y:

hnk-

an,k+ll

-->0 (n-+ oo) (2.1)

k

Sufficiency.Given

:

>0, we can find apositive integer p such that for x m anl for all k>O,

k+p

r

E=k

xr

<L*(x) +

P

+ (2.5)

(Wefixp

throughout

theanalysis).

As

inl.x)rentz’s

proof

(see111; Th. 7)one canshowthat

Z

ankx

_k= v

k+kP

k=O

kO

ank

--4:Tr=

x r

E

nk +"" +

an,k-p

k=p

-

"

an

p-I

+

E

ankxk

k =0

P-11ank

+

an,k_p+l)

x +

E

(5)

SUBLINEAR FUNCTIONALS AND KNOPP’S CORE THEOREH 465 Sincexe m, itfollowsfrom theregularityof

A

that the third and fourthsigmasin(2.6)tend

t)zc,t):,.n-)

,.

Ifwe write

(_n_k.

+ +

an,k-l

k)

l:np="

__

k p

p+

"an

Xk

[F,,pl<-+--

]-

]

hnk+...+an,k_p-(p+l)a,,kllXkl

k=p

_<

I1___11___ Z

.

"lan,k_

r-

ank

p

+

r=O k=p

<p.-i

E

r

Z

"lank-

an,k+l

r=O

k=()

<

P

-b Ilxll y’.

ktnk-a,,k+!

k=(1

(2.7)

SiuccAisstrongly regular, (2.4)holds. Thus theexpressionin(2.7)tendstozero asn---,,,,. 11cnccwet’imlthat

lly (2.5),wchave

I.(Ax)_<_(l.*(x) _c)limsup

,

ttkl

+ Ilxli

iimsul,t

(k,tkl-auk)

timregularityofAand(2.3)wegetthat

L(Ax) < L*(x) + e.

Stut.’ct"inarbitrary, sufficiencyfollows.

’i’111:.()REM.4.I..*A<1.* if andonlyif

A

isF-regularand limsupZ

i+n

n---g-yE

arkl=l

n k r--I

I’ROOF.

Recall that

A

is called

F-regular

if it

maps F,

the classofall almostconverge,t .SCtlUenCes,imo itselfand f-lira

Ax

f-lim x.CorollarytoTheorem 4 in

IOI

givesthenecessarya,d .sufficientconditionsfor

A

tobeF-regular.

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66 C. ORIIAN

f,*(x)

_<

,*(Ax)

_<

L(Ax)

_<L*(x).

Ifxe I:, lhcn

[,*(x)

L*(x) f- lira x. llcncc

l,*(Ax)

L*(Ax) f- lira x. So

A

is i’regular.

"!’

gc

thenecessity of (2.8),we define

(bnk(i))

by

bnk(i)

_ig[

Z.

ark

:1

()bscvc nw flint lhc conditions of

Lcmma

2, l)as_191, arc satisfied.

So

wc

mus

have a I),lcd_ClUCWCysuch thatIlyll and

QB(y)

iimsup sup

Z

Ibnk(i)l

(2.9)

n k

lcnccby (2.9)andF-regularity of

A,

we get

i+n

_

iiminfsup

.

n k

_---vn..arkl

i__i_

i+n

iimsup sup

.

n n+

ark

i+n

limsapsp

(

ark Yk

N l.*(y)N lyN

il _ri

Sullicicncy. Wcf]rst notethat

i+n l.*(Ax) limsupsup

-+-[-

At(x)

n _r=i

limsup sup (---i+n

n

n+-I

E.ark

)xk

weset _i+n

bnk(i)

E.ark

r=l

then

(2.6)

with

ank

relaced by

bnk(i)

holds. Since xe rnand

A

isF-regular, Corollaryto"i’hcorem in

lOi

yields

thatthesecond andthirdsigmaswith

ank replaced

by

bnk(i),

tendtozero as n--oo, uniformlyin i.

On

the otherhand

IFnpl,

with

ank

replaced by

bnk(i)

is notgreaterthan

i+n

P-2

Ilxll

k

Ibnk0)

-" bn,k+

l(i)l

-P

Ilxll

k

I--l--n

+

i

(ark- ar’k+

)1

Since

A

isF-regular, thelastsigmatendstozeroas n.-oo, uniformlyini. lcncc wehave,hy2.5). that

(

i+n

k)

fX

+ +x

p)

L*(Ax)

< iimsup

sup

k k+

(7)

KNOPP’S CORE THEOREM 467

< (L*(x)+c)limsupsup k

Z

Z.

ark

n n+

+ Ilxll

limsUPn

st.pt

_k

(I-!-n

+

I

i

arkl"

--]-

r=E"

ark)

Using

(2.8)

and the fact that

A

i,;_F-regular,weget

L*(Ax) < L*(x) + r.

Sinceeisarbitrary,therequiredconclusionfollows.

Wenowgive anotheringcquality shaqerthanthat of’i’heorcm4.(See"i’hc,,en6 I,clx).Iti,

also ananalogueof Theorem3 given byDevi

!1.

We

firstneedtoproveal.cnma.

LEMMA.5.

Let

(B

(i))

bea

sequence

ofinfinite matrices such thnt(1.3) and(1.5), wih

a,kli

replaced by

bnk(i),

hold."i’hcn,for everyz m_0,we have

B

z

13

y whcrc

and

If, flrther

13

(dnk(i))

(bnk(i)

bn,k+lfi)),

theny--)

O(I))

and z

O(B).

li,n

E

Idnk(i)l

O, n k

uniformlyini,

PROOF.

The first assertion follows from Abel’s partial summation. The consctlucnce of the Result(3.2. I) given by

Duran

I0l.

We

arc now in apositiontogivethei.eqmlitymentioned above. TItEOREM.6.L*A

<

W* if and

only

if

A

isF-regular and

(2.8)

holds.

Beforeprovingthetheorem wenotethat W*iswell-defined(seeDevi

il

!1).

We

now cometothe

proof.

Suppose

thati.*A

<

W*. Since W*< L*,it follows from Theorem 4 that

A

isF-rcgla that (2.8) holds.

Conversely

stnpposcthat

A

isErcgularn(!(2.8)hohls.Ily"i’hcc)rc0l4, we get

u(x) inf

l.*(A(x+z))

_<_W*fx) _(2.1())

ZIll₀

On

theotherhan(I

Nowwrite

i+n

L*(A(x+z))

limsup sup k

-].-[

F.

ark

(’(k

+

Zk)

n _r---i

i4n

bnk(i)

i--]-

.

ark

r.=l

(8)

468 C. ORHAN

Since

A

isF-regular,therequirementof I,cmma5is satisfied. Icnccwchac tl:z-,)(B

S.

t’

get

u(x) i,lf

<l,’(Ax)+t*(Az)=

i.*(Ax) (2

I’I

7m₀

lcncctherequiredconclusitmfollowsfrom(2.!(I) and(2. I).

Thefollowingtheorem is ageneralizationof ResultVilgiven by Kuttncr andM:dhx

I121.

TItEOREM.7.

Let

I1,,1.11

<

and

II:B

II

<

,,,,.

Then

QA(x)

<

qll(X)

ifonly if(13)is a Sclur metht’xland

E

bnk(i) bnk(i)l--

0 (n

--

oo, tmifonrdyin i) k

PROOF.

Sincethe

proof

usesthe technique that

Kuttncr

andM,’ddoxu.,,ed, 2

I.

wc mitthe details.

We

concludethe

paper

withthefollowing remark:Since no Schur mcthd isrcglr.l’hc,ct 7includes theresult that

QA(X)

<

qB(x)

isimpossiblewhen

(I],)

isaregularmethod.For example,

QA(x)

_<

[(x)

(for every

x m),

isimpossible.

REFERENCES

Lorentz, G.G.

A

contributiontothetheory of divergent

sequences. Acta

Math.8__])(194g), 167-190.

2. King,

J.P.

Almost Summable

sequences. Proc. Amer,

Math.

Soc. 17

(1966), 1219-1225. 3. Stieglitz,

M.

Eineverallgen meinerung des Begriffs Fastkonvergenz.

+/-!!.

18_

(1973X

53-70.

4. Cooke,

R.G.

Infinite Matrices

and S.quence

Spaces,

Macmilli:n,1950.

5. Maddox, l.J.

Some

analoguesof

Knopp’s

(’_’ore_theorem. _Intcrnat.J.Matl.[ml b,,l:tt.Sci. 2 979),605-6 4.

Simons,

S.

BanachLimits,infinite matricesand sublincar function:ls. 26 969),640-655.

Rhoades,

R.E. Some

properties oftotally

coregular

matrices. Illinois J. Math.._’1_ {19hi)).

518-525.

8. Schaefer,

P. Core

theoremsfor co-regularmatrices. IllinoisJ.Math.9(I}65),217-21i. 9. l)as,

G.

Sublinear Functionalsand aClass of Conservative Matrices. l].t!i!,_i!!st._l:th.

.c;I.

Sinica. 15 (1987),89-106. I().

11. 12.

Duran,J.P.Infinite Matricesand Almost

Convergence.

Math.Z. 128(1}72),75-

.

l)evi,S.I,. Banachl.,imitsand Infinite Matrices.

J.

l_td_on Math.S_..._i2 (!,76),3)7-,1()1.