Bull. Kyushu Inst. Tech.
(M. & N. S.) No. 27, 1980, pp. 11-16
A NOTE ON LIMIT ORDERS OF OPERATOR IDEALS
By
Ken-ichi MiyAzAKi and Mikio KATo
(Received Oct. 29, 1979)
Introduction For a normed operator ideal [E}r,A] on Banach spaces, the limit order Ai(A, u, v) (1Su, vs: oo) is defined to be the infimum of all Z2O such that
(1) A(I.: l:.l:) s; cna (n == 1, 2,...)
with some constant c= c(u, v, A), where I. is the identity operator from l: into l: (cf. [6], [7]). It has been considerably investigated especially for the ideals [ll., n.], [N., v,], and [llp,,, np,,] of absolutely p-summing, p-nuclear, and absolutely (p, q)-summing operators respectively ([5], [6], [7], [3], [1]), etc.
In this note, we shall define the limit order Zi(A; u, r; v, s) (1 sg u, r, v, sS oo) by using Lorentz spaces l:,, and l:,, instead of l: and l: with a view to studying such an asymptotic property as in (1) of the identity operators between Lorentz sequence spaces. It turns out that the behavior of A,(A;u, r; v, s) is very simple. Indeed, we shall show that Ai(A;
u, r; v, s)=:Zi(A, u, v) for 1s{u, r, v,sf{: oo, which enables us to know, in particular, Zi(np; u, r; v, s) (resp. Zi(np,q; u, r; v, s)) completely (resp. for the most part) since Zi(zp, u, v) (resp. Zi(n.,,, u, v)) is completely (resp. for the most part) known.
1. The space l:,, (1 sg u, rf{g co) is the linear space of all n-tuples 4=:(4i,..., 4.) of real
or complex numbers equipped with
f(:lll ir/u•--i14,1*r)i/r (rÅqoo),
114llu,r :C `=i
kmax {ii/"l4il"; 1giÅq. n} (r :oo),
where {l4il"} is the non-increasing rearrangement of {14il} (cf. [4]). Il • II.,, is a norm (resp.
quasi-norm) if ISrsuSoo (resp. ISuÅqrSoo) ([2]). The identity operator from l:,.
into l:,, is denoted by I.: l:,,--Årl:,,•
LEMMA1([4]). Letls;uiÅqu2f{g oo and lf{ri, r•2Åq- oo. Then, there exists a con- Stant c =c(ui, ri; u2, r2) such that
llln: l:i,ri'l:2,r2llSC (n = 1, 2,•••)•
This is an immediate consequence of Lemma 2.in [4] (see also Proposition 2 in [2]).
We now define the limit order 2i(A; u, r; v, s).
DEFiNiTioN. Let [S}I,A] be a quasi-normed operator ideal on complete quasi- normed spaces (cf. [6], sections 1.1 and 8.1). The limit order Ai(A;u, r; v, s) (lsgu, r, v, sf{ oo) is defined by the infimum of all A2?tO such that
A(In: l:,,.lg,,)gcna (n =1, 2,...) with some constant c=c(u, r, v, s, Z).
If [E}I, A] is a normed operator ideal on. Banach spaces, and if we regard it as a quasi- normed operator ideal on complete quasi-normed spaces, then clearly Ai(A;u, u; v, v) =Zi(A, u, v) holds for 1sg u, vs{ oo.
LEMMA 2 (cf. [3]). For 1S:ul, u2, vi, v2 Sl oo,
11
11
l2I(A,ul,vl)-AI(A,U2,V2)IS ul --" u2 + v, -V2 '
THEoREM. Let lsg u, r, v, s:sg oo. Then, Zi(A;u, r; v, s) ==Ai(A, u, v).
PRooF. Let us first show that Zi(A; u, r; v, s) -Åq Zi(A, u, v). Assume lsguÅq oo and 1Åqvs co. Then, by Lemma 1 for any ui (resp. vi) with uÅquiÅqoo (resp. 1ÅqviÅqv) there exists a constant ci =ci(u, r; ui) (resp. c2=c2(vi; v, s)) such that
II In : l: ,r' l:i II S; Ci (resp. Il I. : l: , . I". ,.Il S; c2) for all positive integers n. Hence A(In: l:,r'l:,s)
S:l ll ln : l:,r --' l:i ll A(In : l:i --' l: i) ll ln : l:i "`' l:,s ll
s;; cic2A(I. : l:, -År l:,) for all n. Therefo re,
Zi(A; u, r; v, s) S Ai(A, ui, vi)
for any ui and vi with uÅquiÅqoo• and 1ÅqviÅqv respectively. Letting ui.u+O and vi.
v-O, we have the desired inequality by Lemma 2. If'u =oo or v=1, we• have Only to' take ui :u=oo or vi= v=1 in the above proof sinoe lll.: las,,.lesll s;I1 and lll.: l7.lr,,ll
S1 for all n.
For the converse inequality, if 1Åqu.f{; co and ISvÅq'co, we take an arbitrary ui
ANote on Limit Orders of Operator ldeals l3
(resp. vi) with 1ÅquiÅqu (resp. vÅqviÅqco), Then, there exists a constant cl=el(ui; u, r) (resp. cS = c6(v, s; vi)) such that
lll. : l"., -År l:,,Il s{g cl
(resp. IlI.: l:,,.I:,11 s{; cS)
for all positive integers n. Hence, A(In: l:i'l:i)
g llIn:'l:i'l:,rIIA(In: l:,r'l: s) IIIn: l"v,s'l"vill
SCICiA(I.: l:,,.I".,,) for all n, from which it follows that
Ai(A, ui, vi) s! Ai(A; u, r; v, s).
Letting ui.u-O and vi.v+O, we have Ai(A, u, v)f{;Ai(A;u, r; v, s). The proof for
u =1 or v = co is similar.
2. In this section, we shall give some results on 7Li(A;u, r; v, s), especially for E}I ==
I7, and ll,,, (the ideals of absolutely p-summing, and (p, q)-summing operators respec- tively), which follows from Theorem and the facts already known for Ai(A, u, v),
Let us recall the definitions of the operators just stated. A linear operator Tbetween quasi-normed spaces E and F is said to be absolutely (p, q)-summing (1 sg qSpÅq co) if there exists a constant p2O such that
n ri
(2 ll TxiIIp)`/p f{gpsup {(2 1Åqxi, aÅrlg)`/q: Ilall S1, aEE'}
i:1 i=1
for any finite family of elements xi,..., x.eE. zp,,(T) denotes infp. I7p,q(E, F) denotes the space of all absolutely (p, q)-summing operators of E into F. In case of p =q, abso- lutely (p, p)-summing operators are called absolutely p-summing, and then I7.,.(E, F) and n,,,(T) are denoted by ll.(E, F) and n,(T).
The class ll.,, of all absolutely (p, q)-summing operators between arbitrary complete quasi-normed spaces (resp. Banach spaces, in particular) is a quasi-normed (resp. normed) operator idgal on complete quasi-normed spaces (resp. Banach spaces) with n.,,.
Since Zi(n,, u, v) is conipletely knoWn ([5], [6], [7]), we obtain the following
CoRgLLARy 1. LetlsgpÅqoo andlsgu, r, v,ssg oo. Then,
(i) in case+of' P :1'
(-I}---l}-+'L- if isu,vs2,
ZI(zl; u, r; v, s)= 1-t if lgu-Åq 2Svsg oo or 2sus co, v2 u',
1
i if 2:{gusg co, vs u',
(ii) in case of1ÅqpÅq2
11
-2'-i+t if ISu,vs2,
1---i; if 1SuS2SvSco or 2SuSp',v2u',
ZI(np; u, r; v, s) = 1
T if 2Susp',vsu'
or p'Su-Åqco,1SvSp, 1
-]b- if p'SuSoo,psvsco,
(iii) in case ofp=:2
11 1
-2---ii"+ir if 1Su,v-Åq2,
1-t if lsus2svsco,
ZI(z2; u, r; v, s) ==
1
i if lSvf{2su-Åq-co, 1
7iE- if 2s;;u,vsg;co,
(iv) in case of2ÅqpÅqco
11
1
-2--'-ii-+T if lsgu,vs2,
1 -h -S if 1 s u sp', 2 -Åqv -Åq co,
zi(z.; u, r; v, s) == i+ (IS7' -4) (;t -" -S-) if p' su sg 2, 2sv sp,
2p '
1
T if 2Su.Åq-co,lsvsp,
1
ir if p' Su s; co,psvs oo,
ANote on Limit Orders of Operator ldeals 15
where 1!p+11p' :1 and 11u+11u' = 1.
Concerning ZJ(np,,, u, v), its behavior is completely known in the cases where 1 == q g ps2 and 2==qspÅqoo ([6], [3], [1]). We accordingly have
CoRoLLARy 2. Let lsg u, r, v, ss co. Then, (i) in case of1sps2
Ai(zp,i; u, r; v, s) =
-!--J!- if oÅq-!-Åq1 --e
os i smax(mS-, tlr+(i--S-)tl,
t-2(1-t)t if ostsS,i+(1-;)tstsl,
t-t't-t if Sstgi,
maxIil-, -lt-[S-+-I;ls-,-i si,
o if $sgii-si,os"stdt+t,
(ii) in case of2spÅqoo
al(np,2; u, r; v, s) =
t if ostsS,ost$+(i--;)t,
t--(1-;)t if OStSt,t+(1-i)tstSl,
S"t't if Sstsi,
max I-lt , -l; -- i}-l s -S- s i,
t+S-t if tstst+t,ostst,
o if -S-+-il)-f{g-l}-si,os-il-s-i----S-•
The other cases where Ai(np,q;u, r; v, s) is known for the most part are omitted
(cf. [3], [1]).
References
[1] B. CARL, B. . MAuREy and J. PuHL, Grenzordnungen von absolut-(r,p)-summierenden Operatoren, Math. Nachr. 82 (1978), 205-218.
[ 2] M. KATo, On Lorentz spaces l.,,{E} , Hiroshima Math. J. 6 (1976), 73-93.
[3] H.K6NiG, GrenzordnungenvonOperatorenideaten(I), Math.Ann.212(1974),51-64.
[4] K. MiyAzAKi, (p, q)-nuclear operators in case ofOÅqpÅq1, Hiroshima Math. J. 6(1976), 555-572.
[5] A. PiETscH, Absolutely p-summing operators in 9,-spaces II, Sem. Goulaouic-Schwartz, Paris, 1970/71.
[6] A.PiETscH, TheoriederOperatorenideale(Zusammenfassung), Jena,1972.
[7] A. PiETscH, Rosenthal's inequality and its application in the theory of operator ideals, Proc. Int.
Conf. on Operator Algebras, Ideals, and Their Applications in Theoretjcal Physics, Leipzig, 1977,
8a88.
Department of Mathematics, Kyushu Institute of Technology
