Some results on von Neumann-Jordan constant for absolute norm (Theory of function spaces and related topics)

概要

72
Some results on von Neumann-Jordan constant for
absolute norm 1
ltq]w~}'I:::k~ · 'i1¥~I~$ .=:it~-

(Ken-Ichi Mitani)

Okayama Prefectural University

fM~::k~ ·

g:j~;W~*

~Jffis!lJJ (Kichi-Suke Saito)
Niigata University

1{-J- o/ J\ ~Fait: B tt Q ~fiif ~Ef9•t!f/{ 0) iuei ~ ~,oz Qt::. th t:ft !7 0) ~M~Et9Ji:'.~iJt
2/f.A ~ :h --C 1,, Q. {~~8~f.t:JE'.~ t L, --C ;{-j- o/ ;\g;gFai t: B tt Q q:it~Ji:'.f!O) )Jx}'I:ra'.if \,, ~
~9 von Neumann-Jordan JE~ (tJ ""f, NJ Ji:'.~) iJ!~ Q. ;::_ O)JE'.~t: J:.-::> --C-~ non-

square't!-?-~.iE:l:JUl~•t!t.t: ~'O)~{iif~Ef9•t!f/{~~{iffi9 Q-:::.

t

iJ!-C~, ~Gt: James;E

~'? modulus of smooothness f.t: ~'O)~{iif~Et9JE'.~t O)f§lLOO~iJ!Wf~~ :h "C\,'Q (e.g.,
[2, 6, 7, 13, 18])
*Wf~-C ti, ~{:;$:8~t.t:1"T o/ ;\g;gFclj t: B tt Q NJ JE'.~t: -::i 1,, --c~~9 Q. ~t:, NJ

JE'.M& t Banach-Mazur fi!eM t 0)00~iJ, G absolute/ Jvb t: Btt Q NJ JE'.M&0)05:\~1§:;z
Q. -:::. 0)~5'! ~ ffl 1, ,--c Day-James g;gFai'? Banas-Fn_}czek g;gFait.t: ~'O) J!{:;$:Et9t.t:g;gFai t: B
tt Q NJ Ji:'.~~ ITTW L,, ~ G t: Banas-Fr<}czek ~ Fai J:: t: B \,' --C NJ Ji:'.~ t characteristic
of convexity t 0) 00~ ~ 1§: X. Q.

Definition 1 ([4]) Let X be a Banach space. The NJ-constant CNJ(X) is the smallest
constant C for which
2_ < llx + Yll 2 + llx - Yll 2 < C
C 2(llxll 2 + IIYll 2 ) holds for all x, y EX not both 0.

Proposition 1 (cf. [7]) (i) 1::; CNJ(X)::; 2 for all Banach spaces X.
(ii) X is a Hilbert space if and only if CN 3 (X) = 1.
(iii) CNJ(Lp) = 22 /min{p,p'}-1, where 1/p + 1/p' = 1, 1::; p::; oo.
(iv) Xis uniformly non-square if and only if CNJ(X) < 2.
(v) CNJ(X) = CN 3 (X*) for all Banach spaces X.

Wtbt:,

absolute J Jvbt:<1::>tJQ NJJE'.~~ITTW9Q. ITTW:15¥!ti [7]-Cl§:x.G:ht::.

NJ Ji:'.M&t Banach-Mazur fi!eMt 0)00{*~ffl\,'Q.
1

Keywords. Day-James space, Banach-Mazur distance, von Neumann-Jordan constant

73
Definition 2 (cf. [14]) For isomorphic Banach spaces X and Y, the Banach-Mazur
distance between X and Y, denoted by d(X, Y), is defined to be the infimum of IITII ·
IIT-1I taken over all bicontinuous linear operators T from X onto Y.
Lemma 1 ( [7]) If X and Y are isomorphic Banach spaces, then

CNJ(X)

d(X, Y) 2

2
:::::;

CNJ(Y) :::::; CNJ(X)d(X, Y) .

In particular, if X and Y are isometric, then

CNJ(X) = CNJ(Y).

Lemma 2 ([7]) Let X = (X, II· II) be a non-trivial Banach space and let
111), where 1 · 111 is an equivalent norm on X satisfying, for a, f3 > 0,

allxll

:S

llxlli :S /3llxll,

X1=

(X,

II·

x EX.

Then

J::. OJ~Ji!! 7'.P G absolute .J Jv b /:. B tt -o NJ ;E'~ 1: ITT~T -o.
Definition 3 A norm

II· I on ffi. 2 is said to be absolute if ll(lxl, IYl)II = ll(x,y)II for any

x,y E ffi..

Theorem 1 ([8]) Let 1 · llx, 1 · IIH be absolute norms on ffi. 2 . Assume that
(i) (ffi- 2 , 1 · IIH) is an inner product space.
(ii) ll(x,y)llx :S ll(x,y)IIH for any (x,y) E ffi. 2 .
(iii) 11(1,0)llx = 11(1,0)IIH and 11(0, l)llx = 11(0, l)IIHThen

CNJ(II · llx) = M2 ,

where M

=

max { :: ~::

(~iEB~OJffliH§) Lemma 2i;p GCNJ(ll ·llx)
1: t IJ , ~ G /:. y = (u, -v) t B
: : :; M2 •

~~ ::; : (x, y) E ffi.

*t:: llxllH =

llx + YII~ + llx - YII~ = M2
2(llxll~ + IIYII~)
.
J::-::, -C

CNJ(II · llx) = M2•

-::. OJ~5'1:t;t:,XOJ J:: -5 1:.-iH~T -o-::. t iJt-c ~ -o.

2,

(x, y) #

(0, 0) }-

Mllxllx > 0t~ -ox =

( u,

v)

74
Theorem 2 ([10]) Let 11 · llx, 11 · IIH be absolute norms on IR2. Assume that
(i) (IR2, 11 · IIH) is an inner product space.
(ii) o:ll(x,y)IIH :S ll(x,y)llx :S ,Bll(x,y)IIH for any (x,y) E IR2 (o:,,B are the best con-

stants).
(iii) In (ii) it satisfies either a II (1, 0) IIH = II (1, 0) llx and o:11 (0, 1) IIH
,BIi (1, 0) IIH = II (1, 0) llx and ,BIi (0, 1) IIH = II (0, 1) llxThen

=

II (0, 1) llx, or

.:. O)JE'.f£-a::ffi1,,-c, Day-James g;gFs5 fp-fq t: B tt Q NJ JE'.~-a::tH~>t Q. Day-James
g;g Fcij t: t, rt QITT. ti Yang G iJ!~ 1J l'l'9 t: fi-::> -c 1,, Q iJ!, 1,, T :h '6 fp-£ 1,fp-£ t,;. t:.' 0) ~ 5JLl
t,;.~i>rt:x-t l,-C-C ib Q ([17, 19, 20, 21]). Theorem 2 -a:: }ti\,' Q .:. t t: J:-::> -C {-:h,J)J'51j..O)
~i'r-'6-fm1:1iib Q iJ!tt•-t Q.:. t iJ!-C ~ Q.
00

Definition 4 (cf. [7]) Let 1 :Sp, q '.S oo. The Day-James

£p-£q

space is the space IR2

with the norm II · llp,q defined by
ll(x,y)llp,q = {ll(x,y)llp,
ll(x,y)llq,

xy 2 0,
xy '.S 0,

where II · IIP is the fp-norm on IR2.
1::; q < P < oo t:X'f'G, IR2 J:O); 1v1,. 11 · llx -a::
ll(x,y)llx = IIT(x,y)llp,q
1
T(x, y) = .,fj,(x - y, x

Lemma 1 J: IJ

Q.

CNJ(fp-fq)

+ y).

= CNJ(II · llx) 1:&"J Q 7'J> G CNJ(II · llx) -a::tt•T:htf'+5:t-Cib

.:_(l)J 1vbt:x-tG-c IR2 J:(l)J 1vb 11 · IIH

-a::

II (x, y) IIH = J22/p-1x2 + 22/q-1y2

(1 :S q < p < oo)

t B -C) -a:: J}.f::. T. J:-::> -C Theorem 2 -a::~fflT Q.:. t 1:!'JZ-a::f~Q.

75
Theorem 3 ([10]) If 1 ::; q::; 2, q::; p < oo and 22/p- 2/q(p - 1) ::; 1, then

C (€ -f ) =
NJ

P

where

q

( (1

22/P(t5 + 22/q-2/p)
+ t 0 )q + (1 - to)q) 2 /q ·

(2 2 /q- 2 /p - t) (1 + t)q-l
+ t )( 1 - t )q- 1

t 0 =sup{tE(0,1): (221 q- 21P

-

In particular, if 1 ::; q ::; p ::; 2, then ( 1) holds.

Corollary 1 ([17, 19, 21]) If either 1 ::; p::; 2, or p > 2 and 22/P- 2(p - 1) ::; 1, then

Remark 1 Let 1 ::; q::; 2, q::; p < oo and 22/p- 2/q(p- l) ::; 1. If His an inner product
space with dimH = 2, then CNJ(fp-fq) = d(fp-fq, H) 2.

Definition 5 ([3, 16]). For;\> 1, the Banas-Frl}czek space IRl is the space IR 2 with
the norm I · I>. defined by
l(x, y)I>. = max { .:\lxl, ll(x, y)lld-

[8] -C~"'- t.:: J::. -5 t::, .: : . O)~Fai Ii frf 1 ~FdJO)--::J0)-®1U~ ctj,.t,ct.:::. t ii-c ~ Q . .::::.
O)@Fai /:: B It Q NJ n:'.~O)f@:!i Yang[16] /:: J::. -::i -C 1~0) J:: -5 t::f~ I? ti -C \,' Q.
Theorem 4 ([16]) For;\> 1,
2

CNJ(IR,\)

=2-

1

;\2 .

.: : . .: : . -C!i Banas-Frl}czek @Fa,~'2i°t'1~0) 21~n.J Jvb@Fai~~A. G, .:::. O)@FdJO) NJ
n:'.~ ~ Theorem 2 ~ J=!h' -C at~T Q.
Definition 6. For a 2". b 2". 1 and 1 ::; p < oo, IR;,b,p is defined as IR 2 with the norm
II . II on IR2 by
ll(x,y)II = max{alxl,blyl, ll(x,y)llp}-

76
1/aP + 1/bP:::; 1 0) t ~ CNJ(ffi.~,b,p) = 2. it-:: a= b = 1 0) t ~ I (x, y)II = ll(x, Y)IIP -Cib
'6'/J>G a> 1, 1/aP + 1/bP > 1 O)~-g-O)cl}.~x.tt!£J::1,,. :=. :=. -c, a 2 b 2 1 r:::xtG, ffi. 2

O),J }VA

11 ·

IIH :a:
ll(x,y)IIH = ll(ax,by)ll2-

t;E'.~9 Q.
p 2 2t
(iii) 0) ~{lf:

9 '6.

II· II, II· IIH Ii absolute -Cib IJ, ~Gt:::: Theorem 2 0) (i),
:a: cl}. t-:: 9. J:: --::> -C Theorem 2 :a:~}=§ 9 '6 :=. t -C {}( :a: 1~ '6 .
J::O)

1

(ii),

1

+ bP > 1.

Theorem 5 ([8]) Let a> 1,a 2 b 2 1 andp 2 2 with aP
p-2

(i) If b::; a(aP -1)2i>, then

(ii) If b > a(aP - 1)

lc__2_
2P ,

then
CNJ(ffi.~,b,p)

=

b2 ( 1 +

G)

1-_2_

-2E_
p-

2

P.

)

In particular, CNJ(ffi.~,b,p) = d(ffi.~,b,P' H)2, where His a two-dimensional inner product
space.

-2E_

2p

Theorem 6 ([9]) Let a> 1, a 2 b 2 1 and p < 2 with ;P +t,, > 1 and aP- 2 +bP- 2
Then
2 ) = 1 + b2( 1 - aP1 )~ •
CNJ (ffi.a,b,p

:::;

1.

In particular, CNJ(ffi.~,b,p) = d(ffi.~,b,p, H)2, where His a two-dimensional inner product
space.
Remark 2 /\7')1 /\~Fl"li X 0) modulus of convexity 8x(c)
ti{}(O)J::-5 t::::;E'.~~t1'6:

t

characteristic of convexity

c0 (X)

8x(c)

=

inf { 1 -

llx; YII : llxll
c 0 (X)

r'elim [12J ti, 9

,oz

=

=

IIYII

= 1,

llx - YII

sup{c E [O, 2] : c5x(c)

=

=

E}

(0:::;

E::; 2),

O}.

-c 0)/\T 'Y /\~rl"!i x i::::xt G-c
1 + co(X)2 < C (X)
4

-

NJ

(2)

77
iJt~ fJ ft-:).:. t ~ ~ l, t::. J:§cO) Theorem 5 t Theorem 6 iJ> G, Banas-Frl}czek ~FaiJ:
t:B1,,-c:;f~~ (2) 0)~%~{1f:iJtf~GttQ ([8, 9]):

<.

a > l, a 2: b 2: 1, a~ + t,; > l t: J1 l, -C X = lR;,b,p t B
(i) p 2: 2 0) t ~ (2) 0)~%~{1f:7'JtJJx;}'z:9 Q.:. t t b :S a(aP - 1)~ !i[P]{[l'l:.
(ii) p < 2 0) t ~, a~ + b~ :S 1 tct_ G /;f (2) 0)~%iJtJJx;ft.

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