THE VON NEUMANN-JORDAN CONSTANT
FOR lpn(Lp)-SPACES
By
Mikio KATo and Ken-ichi MiyAzAKi*)
(Received November 30, 1992)
Introduction
In our recent work [9] the authors gave an interpolation theoretical proof of the generalized Clarkson inequalities ([6] ; see also [7], [10], [8]). The von Neumann-- Jordan constant (for a Banach space) ([5]) is closely related with these inequalities.
It is well-known that Hilbert spaces are characterized as those Banach spaces with this constant 1. 'Besides his celebrated "Clarkson's inequalities" ([2]), J. A. Clark- son showed in [3] that this constant for the Lebesgue space Lp(O, 1) has the value 22max(i/P•i/P')-i. The aim of this paper is, applying our argument in [9] , to show that for general Lp =Lp(X, .X, pt) the von Neumann-Jordan constant for lp"(Lp) (Lp- valued lp"-space) has the above value. The same are obtained for Lp and tp(Lp) as its corollaries by letting n = 1 and n . oo respectively.
Preliminaries
Let Lp==Lp(X, X, pt), 1 ;S P gES oo, be the usual Lp-space on an arbitrary but fix- ed measure space (X, .X, pt). For a Banach space E, let lr"(E), 1$r S oo,be the space of E-valued sequences {xk} of length n with the norm
, II{ Vk }llr(E)=
(,Z"=,llx,Llr)'ir if 1 ,:s .Åq ..,
max 11 xk ll if r= oo.
ISkSn
, The space of all sequences {xk} in E with Il{xk}11r(E)Åqoo is denoted by lr(E), where the norm is as above with n= oo. For E=Lp, we write simply 11 ' llr(p) for ll ' 11r(Lp)
in each case.
The von Neumann-Jordan constant for a Banach space E is defined to be the smallest constant C =C(E) satisfying
*) The authors are panially supported by the Grant-in-Aid for Scientific Research from the
Ministry of Education, Science and Culture (04640172, 1992) .
(i) -23-s llXf("Y.llft,'.11Li!l,ii/,)Y ll2 ;s c
for all xand y in E with llxll2+11yj12;O. As is easily seen, if C is best possible in the right-hand side inequality of (1), then so is 1/C in the left For any Banach space E, 1SC(E) S2; and it is a Hilbert space if and only if C(E)= 1 (Jordan and von Neumann [5]). For E =Lp(O, 1), C(Lp(O, 1))= 22MaX`'iP•'!P""i (Clarkson
[3] ; see also [4], 5. 15).
The von Neumann-Jordan constant for the space lp"(Lp)
We first prepare our tools concerning the complex method of interpolation.
LEMMA 1 ([1], Theorm 5.1.1). Let 1 SPo, PiÅq oo and 1 /P =(1-e) /Po+0/Pi
?vith OÅqeÅq1. Then,
(L p, , Lp ,) [e] = Lp (equal norm s).
LEMMA 2 ([1], Theorems 5.1.2, 4.1.2 and 4.2.1). Let (Xo, Xi) be a compatible co ap le of Ba nach spa ces. Let 1 S Po, Pi S oo (Pi Åq oo ) and 1 /P = (1 - e) /Po + e/Pi
with OÅqeÅq1. Then,
( lp", (Xo), lp", (Xi))[ei=lp"((Xo, Xi) [e]) (egttal norms?.
Let further (Yo, Yi) be another compatible couple of Banach spaces and 1/q=
(1-e)/qo+e/qi (1 $ qo, qi S oo, not qo=qi == oo). Let lpn,(x,) - l,n,(y,)
T:
tp",(Xi) -. I,",(]Yi)
with the norms Mo andMi respectively. Then, T: l7((Xo, Xi)[e]) ' l:((Yo, Yi)[e]) with the norm M satisforing
M$ M,i-eMe.
in the following, iet A = (l -l )
LEMMA 3. Let 1 ;ll P $ 2 and 1/P +1/P' = 1. Then,
(2) llA : lp2(lpn (Lp)). I;t(l,n (L,))11 =2iip'.
PRooF. For the cases where P==1 or 2, (2) is easily calculated as follows:
Letting F, GE lp"(Lp) (P = 1, 2), we have (3) M, = llA : t,2 (l,n (L,)) ---År lk(l,n (L,))ll max (Il F + G lliq), IlF ' G lli(i))
11Fll,,.S +"
IIGPII.. 7Eo 11Flli(i)+11GIIia)
:1
(4) M, == llA : tg (l,n (L,))- lg(tg (L,))ll
(llF + G 11Z,,, + llF - G II3,,,)ii2 = llFll3,,,S+"11GPIIi,,, io (llFl13,,, + IIGII3,,,)ii2
= 2112
by the parallelogram law.
Let now 1 ÅqP Åq 2. Put e= 2/P' (O ÅqeÅq 1). Then, since (Li,L2)[e]=Lp (equal norms)
by Lemma 1, we have by Lemma 2 with Xo= Yo=l?(Li) and Xi= Yi=lg(L2) (li(l?(Li)),l22(IZ(L2))[ei=lp2(lp"(Lp)) (equalnorms)
and
(lk(l?(Li)), l22(l2" (L2))[e] = t3r(lp" (Lp)) (equal norms).
By Lemma2 again with (3) and (4) we obtain
n
