Almost summing mappings

3002 lJu 32 ]AF.tham[ 1v2137030/tham:viXraDANIELPELLEGRINO

Abstract.Weintroduceageneraldefinitionofalmostp-summingmappingsandgiveseveralconcreteexamplesofsuchmappings.Someknownresultsareconsiderablygeneralizedandwepresentvarioussituationsinwhichthespaceofalmostp-summingmultilinearmappingscoincideswiththewholespaceofcontinuousmultilinearmappings.

1.Introduction

Therapiddevelopmentofthetheoryofabsolutelysumminglinearmappingshasleadtothestudyofinnumerousnewclassesofmultilinearmappingsandpolynomi-alsbetweenBanachspaces(see[10],[7],[3],[1]).Recently,Botelho[3]andBotelho-Braunss-Junek[2]introducedtheconceptofalmostp-summingmultilinearmap-pingsandgavethefirstexamplesandpropertiesofsuchmappings.TherecentworkofMatos[8],concerningabsolutelysummingarbitrarymappings,turnsnaturaltoaskwhetheritispossibletofollowthesamelineofthoughtwithalmostp-summingmappings.Inthispaperwewillpresentamoregeneraldefinitionofalmostp-summingmappings,severalexamplesandanaturalversionofaDvoretzky-RogersTheoremforthiskindofapplications.Itwillbeshownthatalmostp-summingmultilinearmappingsaremuchmorecommonthanitwasknownuntilnow.Forexample,weprovethateverycontinuousn-linearmappingfromC(K)×...×C(K)intoaBanachspaceFisalmost2-summing,generalizingarecentresultobtainedin[2].Thispaperalsoanalyzestheconnectionsofalmostp-summingmappingsandtype/cotypeandprovidesvariousexamplesofanalyticalmostp-summingmappings.

2.Absolutelysummingmappings

ThroughoutthispaperE,E1,...,En,FwillstandforBanachspaces.Forp∈[1,∞[,thelinearspaceofallsequences(xj)∞j=1inEsuchthat

∞󰀃(xj)∞j=1󰀃p

=(󰀃󰀃x1

j󰀃p)j=1

p

.

2DANIELPELLEGRINO

Thelinearsubspaceoflwp(E)ofallsequences(xj)∞j=1∈lw

p(E),suchthat

mlim→∞

󰀃(xj)∞j=m󰀃w,p=0

willbedenotedbylup(E).Thesequencesinlup(E)arecalledunconditionallyp-summable.

ThemultilineartheoryofabsolutelysummingmappingswasfirstsketchedbyPietschin[14]andhasbeenbroadlyexplored(see[11],[10],[6]).Thenextdefinitioncanbefoundin[10].

Definition1.AmultilinearmappingT:E1summingif

×...×En→Fisabsolutely(p;q1,...,qn)-(T(x(1)(n)

j,...,xj))∞j=1p(F)

forevery(x(s)

∈labsolutelyj)∞j=1w

(p;q)-summing∈lqs(Es),s=1,...,n.Ann-homogeneouspolynomialP:E→Fisif

(P(xj))∞j=1∈lp(F)

whenever(xj)∞j=1∈lw

q(E).Itisworthobservingthat,inDefinition1,thereisnodifferenceifweconsider

lu

qs

(E)(luq(E))insteadoflw

(E)(lwq(E))(see[10,Proposition2.4]forpolynomials,andthemultilinearcaseisqs

analogous).

Thefollowingwellknowncharacterizationcanbefoundin[4,Theorem1.2(ii)],andissometimesuseful.

Theorem1.LetT:E1statementsareequivalent:

×...×En→Fbeamultilinearmapping.Thefollowing(1)Tisabsolutely(p;q1,...,qn)-summing.

(2)ThereexistsL>0suchthatforeverynaturalkandanyx(l)

j∈El,(2.1)

(󰀃k󰀃T(x(1)

(n)1j,...,xj)󰀃p)j=1

ALMOSTSUMMINGMAPPINGS3

Proof.Sincefisanalyticata,thereareC≥0andc>0suchthat

󰀃1

2

󰀃T󰀃

j=1

n󰀄

λj.

Usingthepolynomialversionofthisresult,itisnothardtoprovethat(see

[12,Theorem4])wheneverFhasfinitecotypeq,everyboundedn-homogeneous(n≥2)polynomialP:E→Fisabsolutely(q;2)-summingand󰀃P󰀃as(q;2)≤

n−2

Cq(F)KG3

k!

df(a)󰀃as(q;2)≤D1Dk󰀃

2D,δa},

∧k

1

m󰀃1

󰀃f(a+xj)−f(a)󰀃q)(j=1

wehavek!k!

df(a)(xj)󰀃q)

∧

k

1

∧k

1

df(a)(xj)󰀃q]

k!

≤

k

df(a)󰀃as(q;2)󰀃(xj)mj=1󰀃w,2

∞󰀃

∧k

D1󰀃(xj)mj=1󰀃w,2

Dk

k=1

p

≤

k,m→∞

ra

limCa󰀃(xj)mj=k󰀃w,p=0

and,bythecompletenessoflp(F),weobtain(f(a+xj)−f(a))∞j=1∈lp(F).󰀁

AnimmediateoutcomeofProposition1isthatDefinition2appliedforn-homogeneouspolynomialsandtheusualdefinitionofabsolutely(p,q)-summing

4DANIELPELLEGRINO

polynomialscoincidesata=0.InordertoprovethatDefinition2forn-linearmappingsactuallygeneralizesthestandarddefinition(Definition1)ofabsolutely(p;q1,...,qn)-summingmultilinearmappingsforq1=...=qn=q,weneedthefollowingLemma,whichisasimpleconsequenceoftheOpenMappingTheorem.Proposition2.Ann-linearmappingTis(p;q,...,q)-summingintheusualsenseif,andonlyif,itisabsolutely(p;q)-summingattheorigininthesenseofDefinition2.

Proof.Consideranabsolutely(p;q)-summing(inthesenseofDefinition2,at

(1)

theorigin)n-linearmapping,T:E1×...×En→F.Then,given(xj)∞j=1∈byProposition1,(T(xj,....,xj))∞j=1∈lp(F).Thus,bytheusualdefinition,itfollowsthatTisabsolutely(p;q,...,q)-summing.

Conversely,consideranabsolutely(p;q,...,q)-summingn-linearmappingTin

(l)(l)

theusualmeaning.Then,ifx1,...,xk∈El,l=1,...,n,wehave

k󰀃1(n)(1)

󰀃(T(xj,...,xj)󰀃p)(j=1

∞uuu

lq(E1),....,(xj)∞j=1∈lq(En),wehave(xj,...,xj)j=1∈lq(E1×....×En).Hence,

(1)

(n)

(n)

(1)

(n)

uuuLemma1.lq(E1×...×En)isisomorphictolq(E1)×....×lq(En).

C

k󰀃1(n)(1)

󰀃(T(xj,...,xj)󰀃p)(j=1

2

k≤C󰀃(xj)kj=1󰀃w,p1...󰀃(xj)j=1󰀃w,pn

(1)(n)

foreverykandanyxjinEl,l=1,...,nandj=1,...,k.Ann-homogeneouspoly-nomialP:E→Fissaidalmostp-summingwhenPisalmost(p,...,p)-summing.

Thespaceofallalmostp-summingpolynomialsisdenotedbyPal,p(nE;F).

∨

(l)

ALMOSTSUMMINGMAPPINGS5

Theorem3.([2,Theorem3.3])For1≤p≤2nandP∈Pal,p(nE;F),thefollowingpropertiesareequivalent:

(i)Pisalmostp-summing.

(ii)Pmapsunconditionallyp-summablesequencesinEintoalmostuncondi-tionallysummablesequencesinF.

ThefollowingdefinitionisanaturalgeneralizationofDefinition3andallowsustogiveexamplesofanalyticalmostp-summingmappings.

Definition4.Amappingf:E→Fissaidtobealmostp-summingata∈EifthereexistCa>0,ǫa>0andra>0suchthat

(󰀁1󰀃󰀃k

(f(a+x1

j)−f(a))rj(t)󰀃2dt)

0

j=1

n!2ne1...enP(e1(a1+x(1)(n)

j)+...+en(an+xj)]−

ei=1󰀃

,−1

−[

1

n!2n

e1...en[P((e1a1+...+enan)+(e1x(1)j+...+enx(n)

j))−

ei=1󰀃

,−1

−P(e1a1+...+enan)].

Forany(x(1)

(n)

j)kj=1,...,(xj)k

j=1,inordertosimplifynotation,wewillwrite

A=(󰀁1󰀃k

(P∨(a1+x(1)

(n)∨1j,...,an+xj)−P(a1,...,an))rj(t)󰀃2dt)0

󰀃

j=1

6DANIELPELLEGRINO

Lemma1assertsthatthereexistsD>0sothat(󰀃(x(1)

(n)

(1)

j)kj=1󰀃w,p+...+󰀃(xj)k

j=1󰀃w,p)≤D󰀃(xj,...,x(n)

j)kj=1foreveryk.Nowsuppose

󰀃w,p

󰀃(x(1)

(n)

j,...,xj)kj=1󰀃1

w,p<

n!2n

a1+...enan)+(e1x(1))

j+...+enx(nj))−

ei=1󰀃

e1...en[P((e1,−1

−P(e1

1a1+...+enan)]rj(t)󰀃2dt)n!2n

(󰀃

e1...en[P((e1a1+...+enan)+(e1x(1)

(n)j+...+enxj))−ei󰀃

=1,−1󰀁10

󰀃k

j=1

−P(e1

1a1+...+en!2n󰀃nan)]rj(t)󰀃2dt)

Ce(1)

1a1

+...+enan+...+ex(n))ki

−1󰀃(e1xj

njj=1󰀃rw,p(e1a1+...+enan)e=1,≤

1

n

󰀃

Ce(1)

(n)

r(ea1

+...+enan)

n!21a1+...+enanD

r

(e1a1+...+enan)

ei=1,−1󰀃(x1

j,...,xj)kj=1󰀃w,p

≤D1󰀃(x(1)

(n)

)kmin{r(e1a1+...+enan)}

j,...,xjj=1󰀃w,p

if󰀃(x(1)

(n)

j,...,xj)kj=1󰀃w,p<δ=

1

ALMOSTSUMMINGMAPPINGS7

foranynaturalk,everyx1,...,xkinEand󰀃(xj)kj=1󰀃w,p<δ,thenfisalmostp-summingata.Proof.

󰀁1󰀃k

1

(f(a+xj)−f(a))rj(t)󰀃2dt)(󰀃

0

j=1

k!

dg(a)󰀃≤Cckforeveryk≥1.

∧

k

Then,foranyboundedlinearfunctionalϕ,definedonF,weobtain

󰀃1

k!

dg(a)󰀃≤Cck󰀃ϕ󰀃foreveryk≥1.

∧k

By(2.2)wehave

󰀃1

2

λkCck󰀃ϕ󰀃foreveryk≥2.

Therefore,definingδaastheradiusofconvergenceofgarounda,ifweassume(xj)mj=1suchthat

󰀃(xj)mj=1󰀃w,2≤δ=min{

1

,δa},3λc)

8DANIELPELLEGRINO

weobtain

m󰀃j=1

∞󰀃1|ϕg(a+xj)−ϕg(a)|≤󰀃

k=2

k−1

dϕg(a)󰀃as(1;2)󰀃(xj)mj=1󰀃w,2∧k

k!

k−2∞󰀃K3G

√≤󰀃(xj)mj=1󰀃w,2

(2k=2

2

foreverycontinuousn-linearmappingT:E×...×E→F.Then,usingtheestimates

ofProposition6,wehave

󰀁1󰀃m

1(n)(1)

T(xj,...,xj)rj(t)󰀃2dt)(󰀃

0

j=1

m

󰀃T󰀃󰀃(xj)mj=1󰀃w,2...󰀃(xj)j=1󰀃w,2

(1)(n)

n!2n

(ei=1,−1),i=1,...,n

󰀃

e1e2...enP(e1x+(e2+...+en)a)

ALMOSTSUMMINGMAPPINGS9

=

n

󰀃

n!2−

n

(n

e2...en[P(x+(e2+...+en)a)−P((e2+...+en)a)])−

(ei=1,−1),i=2,...,n

2

=

󰀁1󰀃k

n

=(󰀃

0

j=1

2

≤≤

n

2

n

2

+

󰀁1󰀃k

1

Qe2...en(−xj)rj(t)󰀃2dt)+(󰀃

0

j=1

n!2n

(ei=1,−1),i=2,...,n

󰀃

(e2+...+en)a

2C(e2+...+en)a󰀃(xj)kj=1󰀃w,p

r

≤D󰀃(xj)kj=1󰀃w,p

min{r(e2...en)a}

for󰀃(xj)kj=1󰀃w,p<δand0<δTheorem4.(Dvoretzky-Rogersforalmostp-summingpolynomials)IfdimE<∞,

thenforp≤2wehave

IfdimE=∞andp>1,thenPal,p(E)(nE;E)=P(nE;E).Themultilinearversionisalsovalid.

Proof.IfdimE<∞,letusconsider{e1,...,en}and{ϕ1,...,ϕn}basisforEandEsothatϕj(ek)=δjk.Givenann-homogeneouspolynomialPfromEintoE,wehave

mm󰀃∨∨󰀃n

ϕj1(x)...ϕjn(x)P(ej1,...,ejn).ϕj(x)ej)=P(x)=P(

′

j=1

j1,...,jn=1

Pal,p(E)(nE;E)=P(nE;E).

Sinceeveryfinitetypen-homogeneousboundedpolynomialisalmostp-summing

(atzero)forp≤2n(see[2,Proposition3.1(ii)]),itisnothardtoprovethatPisalmostp-summingeverywhere,forp≤2.

10DANIELPELLEGRINO

Ontheotherhand,supposethatEisaninfinitedimensionalBanachspace.Itsufficestoconsiderthecase1P(x)=ϕ(x)n−1x.

IfwehadPalmostp-summingeverywhere,wewouldhave,byLemma2,dP(a)almostp-summing(atzero).Sinceϕisalmostp-summingand

dP(a)(x)=(n−1)ϕ(a)n−2ϕ(x)a+ϕ(a)n−1x,

wewouldhaveϕ(a)n−1xalmostp-summing.Sinceϕ(a)=0,wewouldhavethatidEisalmostp-summing,anditisacontradiction.󰀁

Example1.ItisworthobservingthatbyCorollary3,forn≥2,wehavewhereasTheorem4assertsthatPal,2(c0)(nc0;c0)=P(nc0;c0).

Acknowledgment.Thispaperformsaportionoftheauthor’sdoctoralthesis,writtenundersupervisionofProfessorM´arioMatos.TheauthorisindebtedtohimandtoProfessorGeraldoBotelhoforthesuggestions.

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CampusII-CampinaGrande-PB-BrasilE-mailaddress:dmp@dme.ufcg.edu.br

Pal,2(nc0;c0)=P(nc0;c0)