Bull. Kyushu Inst. bl'ech.
(M. & N. S.) No, 24, 1977, pp. 49-57
FACTORABLE OPERATORS THROUGH A DIAGONAL
OPERATOR BETWEEN l,-SPACES
Ken•eichi MiyAzAKi and Mikio KATo (Received October 30, 1976)
Introduction
Factorable operators through a special operator between lp-spaces have recently been studied. One is (p, q)-factorable operators (Terziogle [12], Jarchow [4]), which are factored through the inclusion of l, into l, (pSq). Another is p-factorable operators (Hutton [2]), which are factored through a diagonal D: l..li with 2ge,.,inPMict.-i(D)
Åq co where ct.mi(D) denotes the (n- 1)-th approximation number of D. In this paper we shall introduce and investigate (l,, t,)-r-factorable operators which provide a natural generalization of the latter ones.
An operator T:E-ÅrF is said to be (l,, I,,)-r-factorable (ls{:qÅqpf{co,OÅqrÅqoo) provided T factors through a diagonal D: l,--"l, with 2ee--in'-ict.-i(D)SÅqco (1/s=11q
-- 1/p), and it is shown that such an operator T:E-ÅrF is characterized as an operator of the fo rm T== 2iR, ., i 2.f, O. y. where {.fl,} G IS*)(E'), {y.} E t,•(F) (1 /q + 11q'=: 1) and
Z :, l. 1nrIA,,,is Åq oo ,
1. Preliminaries
Throughout the paper, E, F,... denote Banach spaces, E', F',.,. its dual spaces and .Y7(E, F) the space of bounded linear operators of E into F.
For TEY(E, F) the nth approxjmation number of 7; ct.(T), is defined by ct,(T)
== infII T-All, tbe inf being taken over all AeY(E, F) of rank at most n.
By a diagonal operator Trv{A,} between sequence spaces we mean the operator T({4.}) = {A.e.} where the 7.. are scalars. Without loss of generality we may assume that for diagonals Ttv{A.} we have IZ.1;}rl2.+il for all n. Our discussions are based on the following
LEMMA 1 ([3], [11]). Let 1s!lqÅqp:E{! oo and TN{!1.}: l,-Årl,(IZ.l}irIZ.+il), Then
ct n( T) = {Z :' -n+ 1 lli lS} '/S
where 11s == 11g-1/p.
For 1 Å}-),pf{; co lp(E) denotes the space of weakly ptesummable seq"ences in E, that is,
50 Ken-ichi MiyAzAKi and Mikio KATo
E-valued sequences {x.} such that for each x'G E' {Åqx., x'År} belongs to l., and IS*)(E') the space of all E'-valued sequences {xA} such that for each xeE {Åqx, xhÅr} belongs to
tp•
2e (lp, lg)-r-factorable operators
DEFiNmoN. For lg;qÅqpfsgco, OÅqrÅqco, an operator T:E.F is said to be
(l,, I,)-r-factorabte provided Tfactors through a diagonal D: l.--Årl, having the proper- ty that 2 9,.in'm ' ct. - i(D)S conve rg es (1 ls == 1 lq - 11p).
We denote by ,9ir,,,,,(E, F) the collection of alt (t,, l,)-r-factorable operators and for Te eEZpT ,q;r(E, F) We PUt
fp,q;r( T) == inf{2 ee, .1n'- 1 or.- 1(D)S} i/s,
where the inf is taken over alt factorizations T=VDU, UEY(E, l,), VeY(l,, F)
with Il U 11 :{; 1 , il V II :fll 1 , and Z iP= in'- ` ct. . ,(D)S Åq co .
(t., ti)-r-factorable operators coincide with r-factorable operators (1:fgrÅqco) of Hutton [2],
PRoposiTioN 1. For 1-ÅqqÅqp-Åq oo and OÅqrÅqco, (3`Zcrp',,,,(E, F), f,,,,,) is a quasi- normed space; for any Ti, T2 G -`ZZ'p,q,,(E, F)
(1) fp,q;r(Ti+T2)SK(P, q, r) [fp,g;r(Ti)+fp,q;r(T2)]
where K(p, q, r) ==2i+ifP+ilq+max(tls,"ls) (1/s=11q-11p).
PRooF. Let TiG-gZi;.',,,,(E, I7) (i=l,2), Then, for any sÅrO there exist factoriza-
tionf Ti = ViDiUi g. uch that Il Uill s{ 1, Il Vill s;; 1 and
{Z ee.. i n'- i ct.. i(Di)S} iIS Åq f.,,,,( Ti) + e12.
Define I: E-ÅrE{?E, S: Fg?F-F, R: l,{? t,.l. and J: l,-År t,9? l, as follows:
Ix == (x, x), S(x, y)=x+y,
R({4`.}, {q,}) ==(s:i, nyi, g2, n2,•••), J({4.})=({42n-i}, {g2n})'
Let D.v{Ai,pti,Z2,pt2,.,.} where DiN{Z.},D2N{pt,}. Then we have the fo11owing commutative diagram
Factorable Operators Through a Diagonal Operator BetweÅën tp-Spaces 51
E -!l-!,-Å}nt!z-,+72 F
il ls
E{?Ei,;esi;•eT, Fg?F
uleu2i lvlev2
lp S) lp -ii;i-ei55;"'eD2 lq {I? lq
Rl IJ lp T lg
If we put A=2"IPR(Ui(EDU2)I, B=2-iS(V,G)V2)J and Do=2t'`IPD, then we have
T== BDoA, llAH s{ 1 and ll Bll s{ 1 since 11RH -Åq 2'/p and "SII :E{; 2. We now show that 2ge,=in'-ict.-i(Do)S converges, Since D=J-`(DieD2)R-i, ilJ-i11g2i/q and llR-ill s{;1, we have
ctn(Do) == 2 l ' 1 IPct.(D) :f{; 2i +ilp+ifgct.(Di ({DD2) •
In case of 1 .Åq rÅq co,
{2 ee,=in'- i ct.- i (Di (D D2)s} i /s
== {Åí: .. o(2n + 1 )'- ' ct2.(Di (il) D2)S + 2 ee,.i (2n)'- i ct2,- ,(D iOD2 )s} iis
f{; {Z:=o(2n + 2)"- ` ct2.(DieD2)S + 2 ee. i (2n)'pt i ct2.- 2(Di OD,)s} ' is =: {22:. i (2n)'- t ct2(. - i )(Di Åq{D D2)s} i ls
f{;l 2'fS{Zee.. in'ny i max ct.. i(Di)s} ils
i
:s{ 2'1s{2 R, ., in'" i (ct.- i(Di) + ct." i(D2))S} i/S
S 2'!S[{2:., in'- ' ct.- i(Di)S} iIS + {2 ee. in'- i ct.- i(D2)S} iIS]
Åq 2'IS Vb,q;r( Ti ) +fp,q;r( T2) + 6] '
Hence we have
{2:.. i n"- i ct. - i(Do)S} i /S s{; 2i ' iIP' i 1q"IS[fb,,,,( Ti) +fp,g,,( T2) + e] ,
which jmplies that 7ii + T2 G ,s%t ,,,,(E, F) and (1) for 1 s{I rÅq oo , In case of OÅq rÅq 1,
{Z =inr"ict,.uzi(Di(DD2)s}tls
= { Z) :]. o(2n + 1 )'n ' ct2.(Di ({D D2)S + Z ee. i (2n)'- i ct2,, - t(Di ({D D2)S} ifS
52 . Ken-ichi MiyAzAKi and Mikio KATo
f{; {Z ee. o(2n + 1)'m ' ct2.(Di e)D2)S + 2 .co. i (2n --- 1)'-- i ct2.m 2(Di eD2)S} i /S
= 2i ls{Z 9,.i(2n - 1)'- ' ct2(.. i)(Di (D D2)S} `/S s; 2ils{Z ee= in'- i(ct.m i(Di) + ct. -. i(D2))S} '/S
s{ 2' /S[{ 2 ee= i n'- ' ct. - i(D i)S} i IS + {2 ee. i n'- i ct. - i (D2)S} 'IS]
Åq 2i/S[f,,q;r( Ti) +fp,q;r( T2)+ 8] '
Hence we have
{2:. 1 n"- 1 ct. - 1 (Do)S} 1 /S S 21 ' 1 /P" 11q" 1 IS[fp,,,,( Tl ) +f,,,,,( T2) + e] ,
from which it follows that Ti+T2e 20i',,,,,(E, F) and (1) for OÅqrÅq1. The other parts of the proof are easy.
PRoposmoN 2. Let 1 :E{g qÅqp s;l co, OÅq rÅq co, Let RGY(E, F), Te .9 ,T ,,,,(F, G) and S6Y(G, H). Then STR G ,E9tl,,,,(E, H) and
(2) fp,q;r(S TR) Sll ll Sll fp,q;r( T) llRII'
PRooF. For any sÅrO there exists a factorization T=:VDU, UG.f2e(E, l,),
VEY(l,, F) with IIUil mÅq1, IIVII sl1, and
{2:= i n"" ct.. i (D)'S} i /S Åq f,,,,,( T) + e,
Let Wi = 11 11 RH UR, W2 = 1lll SUSV and Do = IISII llRll D. Then T= W2 Do lxVi, II Wi ll S l,
ll lxV2 ll sg 1 and
fp,q;r(STR) S; {2 ff,L i n'- ' ct.. i(Do)S} iIS -Åq Il S11 {2: ,,. i n'- i ct,m i(D)s} i/s ll R II ff{i llSII (f,,,,,(T) + e) llRll ,
from which it follows that STR G ,29.i ,,,,.(E, H) and (2).
3. Tensor product representation of factorable operators
PRoposiTioN 3. Let 1 f{; qÅqp fE{ co . An opera tor T: E.F fa ctors through a diago- nal D: lp-Årl, if and only i,f Thas a representation of the form T=::,l,.iA..L,O. y. such that {1.}E l,,, {f.}e iS")(E') ancl {.y,} G l,•(I7) where 11s=11q-1!p and 11q+11q'==1.
PRooF. Let T= VDU where DFw{A,}: l,-.l,, UeY(E, l,) and VGY(l,, F). De-
Q,
fine f.GE' by Åq.x;,L,År =:-=- ÅqUx, e.År and pUt.y. =Ve. where e.=:(O,-.., O, 1, O,,..), Then,
Factorable Operators Through a Diagonal Operator Between t.-Spaces 53
we have {f.} E tS")(E'), {y.} E l,t(F) and for every xGE
Tx = VD Ux
= V(2 ifl- i An ÅqX, .fh ' en) = 2:-iZnÅqX, fhÅrYn'
Let T=]IE iP..i2.f.(g)y. where {1.}Gl., {f.}GIS")(E') and {y.}El,•(F). Define UEY(E, l,) and VGY(l,, I7) by Ux={Åqx,.fhÅr} and V({4.}) :2",=ie.y. respectively, Let Dtv{7..}; l..l,. Then, we have for every xEE
TX =2:=iAnÅqX, .f)iÅrYn = V({An Åq X, fh ' })
=VDUx,
LEMMA 2. Let OÅqrÅqoo, Thenforany positive i,nteger n, -L n" :s{ 2:•'a ii"-' :f: n' (1 f{; rÅq co) ,
r
HTL nr ;}) 2:i. ,i"" i ;!i n' (OÅqrÅq 1) .
r
The proof is easy.
THEoREM 1, Let 1 f{l cl ÅqpS co, OÅq rÅq co . ttln ope ra tor T: E--ÅrF is (tp, lq )- r- .factorable if' and only if' T has a representation oj' the fbrm T= 2ee,=i2..f],(DÅr?. where
{f,} e tS")(E'), {.v.} e I,•(F) and Z:.in'IA.ISÅq oo (11s= 1/q- llp),
PRooF, By virtue of Proposition 3 we have only to prove that 2:.in'-ict.-i(D)s
Åq co if and only if Z iR,=i n'1 2..Is Åq oo for a di agonal DN {Z.} : l, •-År l, (l ls = 11q -- 1/p). By
Lemma 1,
2:= i n'm i ct.- i(D)S : Z ee, ., i n'- '(2 ee•..1lilS)
: 2if)- i(1Åí Y• .. ii""i)1A.1s.
Consequently, by Lemma 2 we have
(3) - 2il,l. i n' 1 2,, lss :i?= in'm i ct .mi(D)S Åqm 27rL. in" l Z. IS for 1 s: rÅq co
r
and
(4) -•-;-- 2iR,.tn'l ,Z,ISÅr- :if,l,.inr-ict.Fi(D)'S ;}li 2:,il. inr1Z.IS for OÅqrÅq1.
54 Ken-ichi MiyAzAKi and Mikio KA'ro
This completes the proof.
4. Inclusion relations
PRoposiTioN4. (i) Let lff{qÅqpif{:pf{;co, OÅqrÅqoo. Then, ,y".,,,,(E,F)
c ge,,,,,,(E, F) and fot" eaCh TE e2%t ,q;r(E, F) fpi,q;r( T) Sg MaX (1 , r' /P i- i /")fb,,,,(T) .
(ii) Let ls;qSqiÅqpSoo, OÅqrÅqco. Then, 3ut,,,,,(E, F)c,taiCe'.,,,,,(E, F) and fbr each
TG cf9tiii,,q;r(E, F)
f,,,,,.( T) g max (1, ri1qm i/qi)fb,,,,(T) .
(iii) Let 1-ÅqgÅqp:E{g oo, OÅqrif{;rÅqco. Then, .2`kff',,,,,(E, F)c.ESt ,,,,,(E, F) and lbr each
TG cS/i.iii,,q;r(E, F)
fb,q;ri(T) S{ fb,q;r(T) •
PRooF. (i) Let TG y4i,1,,,,,(E, F). Then, for any ÅíÅrO there exists a factorization T= VDU such that IlUII :f{; 1, IIVII f{gl 1, D-{2..} and
{2:,. in'- i ct." i(D)S} iIS Åq f,,q,,( T) + e
where l/s----=11q-11p, Let Ditw{Zsuo} and D2N{AÅrg/Si} where 1/po==I!pi-11p, 1/si==11q--11pi. Then we have the following commutative diagram
E-m-m7..-h.,,' F
ui lv
lp r---D=-'--' lq
DN /.,
lpi
If we put Uo=1111DiI]DtU, Do=IIDillD2, we have T== VDoUo, IIUolls1 and
(5) {2 il,2,. in'-j ct,-i(D o)Si] '!Si = ll D, ll {2 :,2. ,n'- '(2 r• V=,IAils)}ilst
= IIDi Il {2:.,, in"- i or.-,(D)s}iisi SLIIDi ll {fp,q;r(7') + e}Sl"' L
'I'hk imp. Iies TE pt,,,,,,(E, iF). On the other hand, we have by (3) and (4) (6) IIDi ll g{ ]Åí i, ny il?.,ls} iiPo
Factorable Operators Through a Diagonal Operator Between tp-Spaces 55
S; {2 :, l= in'IZ.ls} ifpo
S max (1, ri/"o) {Z:= in'- i ct.- i(D)s} i/Po g max (1, r'fPo) {f,,,,,( T) +e}s/Po.
Since slsi+slpo :1, it follows from (5) and (6) that fpi,q;r( T) K MaX (1, r'IPO) {f,,,,,( T) +e} . Hence we obtain the desired inequality.
(ii) can be proved in a similar way. (iii) is trivial.
We now recalt the definition of the space l.,,.
For 1 f{: p, q f{: oo l,,, is the space of scalar O-sequences {e.} such that
II {4,}u,,,=i (2:=inqf"-il4.l"q)'iq (q Åq ,o), ( supn'IP14.1"
(q =oo)
is finite, where {ie.l"} is the non-increasing rearrangement of {l4.l} (cf. [6], [5], [7]).
It is well known that l,,,clp,,, if pSpi, and lp,qclp,q, if q:E{gqi•
ExAMpLEs. (i) Let 1:E{qÅqpiÅqpSco and s/(si-s):fgrÅqoo where 11s=11q-11p, 1!si =11q-11pi. Then, there exists an operator which is (t,,, t,)-r-factorable, but not
(l,, l,)-r-factorable.
(ii) Let IKqÅqqi Åqps co and sl(si-s) SrÅq co where 11s = 11q -1!p, 11si = 11qi
-- 1!p. Then, there exists an operator which is (t,, l,,)-r-factorable, but not (Ip, l,)-r•- factorable.
(iii) Let ISqÅqp:E{;oo,OÅqriÅqrÅqoo. Then, there is an operator which is
(t,, t,)-ri-factorable, but not (tp, l,)-r-factorable.
PRooF. (i) Since stl(r+1)s;slr and sÅqsi, we can get a {A.}G t,,i(,+o,stXts!,,s (e•g•, A. =n"'1'[log(n+1)]-`/s). Let TN{Z.}:l,,---Årl,. Then, since 2ee.in'i2.ISiÅqoo, it follows from Theorem 1 that T is (l,,, l,)-r-factorable. If T were (l., l,)-r-factorable, 29.in'-ict.-i(T)S must be convergent. However, we have by Lemma 1
2 ee., in'-` ct.- i(T)S=: :Ii: ge. in'-i(Z:..IZi1si)slsi !}ir 2 9,,. in'-' i l7,,,ls
= co}
which is a contradiction.
(ii) We take {A,}•El,,1(,+i),,,Xt,i,,,. Let TN{Z.}: t,--.t,,. Then, it is shown in a similar way that Tis (tp, lg,)•mr-factorablc, but not (t,, i,)-r-factorable,
5.6 Ken-ichi MiyAzAKi and Mikio KA'ro
(iii) We take Vw.}Gt./(,,+i),,Xl.f(,+i),, (e,g., 7..'==n'("'i)ls[log(n+1)]-i/s). Let
TrNv{?v.}: l.-t,. Then, since 2ff.,in'i17..ISÅqoo, it foHows from Theorem 1 that T is (t,, l,)-ri-factorable, On the other hand, we have by (3) and (4)
2 9, ., i n" ict ,,mi(T)S 2}i min (1, ; ) 2:=i nr IAls=: oo ,
which implies that Tis not (l,, t,)-r-factorable,
5. Relations between factorable operators and some other operators We recall the definitions of operators of type l, and (p, q, r)-nuclear operators.
For OÅqpÅq oo an operator T:E.F is said to be of type l, (cf. [9]) if the sequence {ct.(T)} belongs to l,.
In [10], an operator T: E.F is said to be (p, q, r)-nuclear, OÅqp, q, rg oo, 1!p+1!q + 1!r2 1, if Thas a representation of the form
TX = E] Y,Li2nÅq-X,f]iÅrJ"n (VXGE), or symbolically
T= 2 ee= , A.fh (g) y,i, where {7..} G l., {fh} G IQ*)(E') and {.v.] G I,(F).
It should be noted that (p, oo, p')-nuclear operators (1 f{;pÅq eo, 11p+1!p'= 1) coin- cide with p-nuclear operators in the sense of Persson and Pietsch [8], and (p, oo, oo)- nuclear operators (OÅqp:s{; 1) are exactly p-nuclear operators in the sense of Ha [1].
THEoREM 2. Let 1 :f{: qÅqp -Åq oo and OÅqrÅq co. Th en,
(i) Ever.y (l,, l,)-r-Lfactorabte operator is ojf' t.vpe l,1,+,,fbr alt eÅrO (1/,s=1/q- llp), (ji) Ever.v operator o:f= t.ype ti!(,+i) is (l,, l,)-r-•Lfactorable.
PRooF. (i) Let T: E.F be an (t,, t,)-r-factorable operator. Then, we have
2 C.".. i nr- i ct. . i( T)S Åq co , or {or.- i(T)} G l,!,,,. Consequently, {ct.- i(T)} E l,1,+, for every 6ÅrO since l,,,cl,,,,, for OÅqpÅqpiS oo, OÅqq, qiK co (cf. [6], [7]).
(ii) Let T:E.F be an operator of type til(,+i). Then, by Proposition 8,4.2 in
[9] T is represented in the fo rm T = 2 :, l.. i A. J), (2{) y. where {2.} E ti !(,+ i), ll .fl. II g 1 and [b'. II
:fi{ll for all n. Since Ii!(,+i)clif(,+i),i, we have 2ee,=inrlZ.lÅqco. Therefore it follows from Theorem 1 that T is (l,., ti)-r-factorable, Coiisequently, by Proposition 4 T is
(tp, lq)-r-factorable.
PRoposiTioN 5. (i) Let 1sgqÅqpsl co and OÅqrÅqco. Then, every (t,, t,)-re•factora- hte operator i,s (t, p, q')-nttdear/br alt t with 11(r+1)ÅqtSs where 11s---1/q-1!p ancl 1lq+1lq'= 1,
Factorablc Operators Through a Diagonal Operator Between l.-Spaces 57
(ii) Let 1:E{qÅqco and OÅqrÅqoo. Then, every (l., l,)-r:factorable operator is t-nuelearfbr all t with q/(r+1)Åqts{ q.
PRooF. (i) is an immediate consequence of Theorem 1 and the fact that lp,qclp,,,i, for OÅqpÅqpi -Åq co, OÅqq, giS oo.
(ii) Let T: E.F be an (l., l,)-r-factorable operator. Then, by (i) Tis (t, oo, q')- nuclear. for each t with q/(r+1)Åqts{q. Therefore, T is written in the form T==
2) :,2- i Z. .fh (g) .y. where {2.} E l,, ll fh ll :{: 1 (Vn) and {y.} G t,• (F). Consequently, in case of
q/(r+1)Åq1, if q/(r+l)Åq1gtgq, Tis t-nuclear in the sense of Persson and Pietsch since {y.}Gl•,•(F)(11t+ilt'=1), and if q/(r+1)Åqtf{{lf{{q, Tis t•-nuclear in the sense of Ha since lly.llg1(Vn). In case of q/(r+1)21, similarly we have Tis t-nuclear in the sense of Persson and P. ietsch for each t with ql(r+1)Åqtgq.
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Department qf Mathematics, Kyushu Institute of Technology
