WŁODZIMIERZ ODYNIEC WSP w B y d g o s z e z y
ON UNIQUENESS OF MINIMAL PROJECTION I N 11
B l a t t e r and Chen ey [”l } s t u d i e d t h e e x i s t e n c e o f a m i n i m a l p r o j e c t i o n f r o m t h e Banach s p a c e 1^ o f a l l a b s o l u t e l y hum m a b le r e a l s e q u e n c e s o n t o i t s c o n d i m e n s i o n one s u b s p a c e D = f ^ ( 0 ) , w h e r e f € I n p a r t i c u l a r , t h e y p r o v e d t h a t i f t h e l i n e a r f u n c t i o n a l f i s o f t h e f o r m : ( O f = ( f n > ’ 1 = f , > f 2 ^ . . . ? 0 , f 3 ^ 0, Th en t h e h y p e r p l a n e f ” ^ ( 0 ) i n t h e s p a c e 1^ a d m i t s a t l e a s t o n e m i n i m a l p r o j e c t i o n , whose norm i s g r e a t e r t h a n 1. I n t h e p r e s e n t p a p e r we e s t a b l i s h a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e u n i q u e n c e s s o f a m i n i m a l p r o j e c t i o n f r o m 1^ o n t o f 1 ( 0 ) u n d e r t h e a s s u m p t i o n ( 1 ) . Our t e r m i n o l o g y and n o t a t i o n f o l l o w s [ l ] and
[23
. L e t B be a Banach s p a c e o v e r t h e f i e l d R o f r e a l n u m b e r s , SR = [ i £ B : ||x||:l} , t h e u n i t s p h e r e . By B* we d e n o t e t h e d u a l o f B, F o r e v e r y f £ B * \ ^ 0 ^ we s h a l l c o n s i d e r t h e f a m i l y o f l i n e a r o p e r a t o r s ^ 2 ^ Pf , z = 1 “ f ® z : B —* B, z £ B, i . e . P - _ ( x ) = x - f < x ) z . I , z L e t z £ f 1 ( 1 ) , t h e n P „ I , z i s a p r o j e c t i o n f r o m B o n t o D = f “ 1 ( 0 ) , ( c f . [1 3 ) . L e t ( 3 ) q ( f ) = i n f {l| P f f Z l|: z £ f ~ 1 ( 1 ) } , t h e r e l a t i v e p r o j e c t i o n c o n s t a n t o f t h e p a i r ( B, f ” V o ) ) ,6 and l e t
O )
Gf = { z £ f_1
C1
) :l|Pf > z ll
=q C f > ) ,
t h e s e t o f t h e p o i n t s z c o r r e s p o n d i n g t o m i n i m a l p r o j e c t i o n s . Th e e q u a l i t y c a r d ( G^.1= 1 e x p r e s s e s t h e u n i q u e n e s s o f a m i n i m a l p r o j e c t i o n on f “1
(0
) . F o r e v e r yz =Cz i )
we w r i t e z >0
( r e a p . z ? o ) i f f z^ >0
/ r e a p . z ^0
) f o r a l l i n d i c e s i . We l e t B+ = [ z 6 1 1 : * ?> ° } • Ve s h a l l n e e d t h e f o l l o w i n g r e s u l t s o f h e p a p e r s [1
] and[23
: P r o p o s i t i o n Lenina L e t B = 1^ . I f f S g « s a t i s f i e s t h e c o n d i t i o n (1
) , t h e n( 5 )
of =
: * * o , llPf > z H =
.
P r o p o s i t i o n 2 . ( f l ] , Lemma 3 ) . L e t B = 1 , f € . S « , and Th en (6
) l|pf > z l l = » u p J h - f n z n | ♦ | f n | (||zn - l z n D ) . I n p a r t i c u l a r , i f f>0
and z0
, t h e n ( 7 ) l l pf | Z l l = 1 + S« P r i * w h e r e r ± = ( || z II -2
z ± ) f i . P r o p o s i t i o n 3 . f [ l 3 . T h eo rem7 ) .
L e tB
= 1^ and l e t f S g * s a t i s f y i n g(1
) . Assume1/0
= oo,0
• ° ° =0
andd e n o t e ^ j
<8
) ‘ j '1
? , f i ’ b J = * ? ’ f ‘ ' ’ P j = V « - 2 > f o rJ > 2
;
( 9 ) C j = min S / j b j . , , a j_1
} f o r J » 2 ; (10
) k = k ( f ) = max ^ j : C j - j ^ -3
} . T h e n ( i ) i f f = f ° > , t h e n q ( t ) =2
; ( i i ) i f f / f ° ( t h e nq ( f ) = 1
+ u , w h e r e- l / f kU k -
2
) + i f a ^ k-2
, (11
) u = I U ( V k - k) -1
, i f a ^ k-2
and 0 2 ) (3k ^ 1/ f k • Remark 1. By ^2 3 , Lemma 3 . 1 » R C f) ^ 2
f o r f / f ° . A d a p t i n g P r o p o s i t i o n3
t o t h e c a s e o f t h e n - d i m e n s i o n a l s p a c e B = l ” ( n > h ) we g e t : i f 1 = f1
^ ^ f ^ & 0 , f ^ ^ O , t h e n q ( f ) =1
+ u w i t h u d e f i n e d b y(1
1
) , ( o f . CZ-1and
C31 ) , P r o p o s i t i o n h . ( [ 2 3, T h e o re m 3 » 2 ) . L e t B = l ” , n-^.3, f £ S Q* s a t i s f y i n g (13
)1
= f3
> o . L e t ( * ■ ( * ) > ^ »(
1
^) m = m (f ) =
J
C » i n £ i e { 3 , ^ ... k ( f ) > l / f i + 1 = f t c ( f ) 5 > o t h e r w i s e . _/J Th en t h e m i n i m a l p r o j e c t i o n o f B o n t o f ( 0 ) i s u n i q u e i f f one o f t h e f o l l o w i n g two c o n d i t i o n s i s s a t i s f i e d : , C i ) a > ni-2
; m ( i i ) ani< m -2
, f2
<1
, am_1
> m-3
. P r o p o s i t i o n 5 . £ f 2 3 . T h e o re m 3 - 3 ) * L e t B = 1™ , b , f S g * s a t i s f y i n g (1 3 3 , m = m ( f )5
3
, S = l ” , f = ( f , . . . , f m ) €. B * . T h e n m ( f ) = m, q ( f J = q ( f ) and G f = [ z e . B + : ( n.^ , . . . t x m) e G ^ » 2 j =0
f o r m < r j < n3
; i n p a r t i c u l a r , c a r d ( Gf ) = c a r d ( G^r). T h e n e x t t h e o r e m i s o u r main r e s u l t : T h e o r e m 1 . L e t B = 1^ , f t S g * su c h t h a t 1 = f5
f ^ f ^ >0
. Then ( i 3 i f f = f ° * (1
,1
, . . . ,1
, . . . ) t h e n c a r d ( Gf ) >1
; ( i i ) i f f 4 f ° ,8
=1
™ and f = ( f l f . . . , f m) , w h e r e /8 \ m i s g i v e n toy ( 1 1*) , t h e n t h e m i n i m a l p r o j e c t i o n o f B o n t o f ( 0 ) i s u n i q u e i f f t h e m i n i m a l p r o j e c t i o n o f B o n t o f ( 0 ) i s u n i q u e . P r o o f , t i l L e t f = f ° and z 1 = ( 1 / 2 , 1 / k , i / ' t , 0 , 0 , . . J » e B , z 2 = 0 / 3 . 1 / 3 , 1/3, 0 , 0 , . . .") € .B . E v i d e n t l y z 1 , z 2€ f - 1 f 1 ) , Pf o z 1 / Pf 0 z 2 . By ( . 7 0 P r o p o s i t i o n 3, and Remark 1 , ||P o lj|=l|P o 2 II = = 2 . Hence c a r d ( G■ ^)>1. f , z f , z 1 ( i i ) L e t f / f ° . By Remark 1, q ( f ) = 1 + u ^ 2 . L e t m = m ( f ) b e d e f i n e d b y ( 1 * 0 . Le t B = 1™ , f = ( f 1 , . . . , ) . R e c a l l , t h a t b y P r o p o s i t i o n 5 , q ( f ) = q C O * F i r s t , we p r o v e t h a t ( 1 5 ) Gf c { z £ B t : ( z , . . . ,z hV € G - , z ± = 0 f o r a l l i > m^. To t h i s en d l e t > z e. Gf and z = ( z 1 , . . . , z m ) e B. By ( 5 ) , z £ b . + I n t h e c a s e f . = 0 ( a n d h e n c e f , = 0 f o r a l l j^-m +1 1 ~ _ i m+1 j we h a v e z <=-f~ ( 1 ) , a n d , b y P r o p o s i t i o n 2 , q ( f II ~||= 1 + max ( H z
II
- Z x ^ ) f ^ : 1 i i i ^ 1 + max { ( I I z II - 2 z ^ ) f ^ : 1 i i f n j = q ( f ) . T h e r e f o r e II z II = l l z ll and h e n c e z . = 0 f o r a l l i ? m+1 , and z 6. G£. Now c o n s i d e r t h e c a s e : f / 0 . O b s e r v e t h a t m+1 (16
) z 1 > 0. I n d e e d , llz||=||zll | l f | | ? f ( z ) = 1 and b y P r o p o s i t i o n 2 and Remark 1, II z ll - 2 r. ^ = (| lz II - 2 z { ) f 1 — H z H ” 1 = u -d 1 . By ( 1 6 ) and t h e f a c t t h a t z €. B+ , we h a v e t -= + * * * + f z > 0 . E v i d e n t l y y = t - 1 z ^ fV
1 ) . m m By P r o p o s i t i o n 2 t h e r e e x i s t s an i n d e x i o s u c h t h a t 1 i - - m andII
P;r ~ H = 1 + ( ll y ll - 2 y ) f . Hen ce ° * * y o ot u ^ t ( | | p - ~ | ) - l ) = t(||y ll - 2 Y i ) f t = ( ||ty II - 2 * lo) f i o = ^
= ( I . « -
±
* J - Vt.°-
V 11* 11 - » • , ) -r
, * j= m + 1 ° o o o o o cx? ' u - o z n * 3 = m + i i . e . i= e ( f ) t 2 ^ G£» T h i s e s t a b l i s h e s ( 1 5 ) . F i n a l l y we p r o v e t h a t (1 7 ) G j ^ |z 6 B+ ! Cz -j» • • • i z ra) = 0 f o r a l l i ^ m+11# T o t h i s en d l e t z ^ G - , z = ( z , . . . , z ) , and 2 = CZ 1 » •• • » z mi ° » • • • ) <£■ B. By P r o p o s i t i o n 5 z e.B+ and t h e r e f o r e z 6 ®+ * P r o p o s i t i o n 2 and b y ( 1 ) l|Pf > J l = 1 + max ( u , f m+1 !l z l l !j , I n t h e c a s e : f = 0 , b y P r o p o s i t i o n s 3 and 5, we h a v e 1114- I llPf | Z l l = II = q ( f ) = q ( f ) and z £ Gf . I n t h e c a s e w h e r e fm+1 > ° » we l e t f = C f 1 * • • • * f m+1) » B = l “ +1 , z = ( ^ 1 i . • • , z m, 0 ) £ B . By P r o p o s i t i o n 5 , Gf = ( X € B + V > * Gf ’ * m+1 = 0 } and l l P f ^ M l P f ^ l l = q ( . f ) = q ( f . ) = 1 + u . H ence f m+1HS|U u , b e c a u s e H P £ g I! = 1 + max ^ u , f m+1 |l zl|} = 1 + u . O b s e r v e n e x t t h a t
II
zll = R %II
= ||zl|. T h e r e f o r e II P f 2 II = 1 + majc f u » f m+1
H Z H } = 1 + u = q t f ) and z £ G f . E x a m ple 1 . L e t f = ( 2 1 “ n ) e ( l ) * . By (1 0 ) a i . u ( l U ) , m ( f j = = k ( f ) = 3. H e n c e , b y T h eo rem 1 and P r o p o s i t i o n U, t h e m i n i m a l p r o j e c t i o n f r o m 1^ o n t o f ~ 1 ( 0 ) i s u n i q u e . Remark 2 . S i n c e t h e s p a c e 1^ i s s y m m e t r i c , o u r T h e o re m 1 y i e l d s a n e c e s s a r y and s u f f i c i e n c o n d i t i o n f o r t h e u n i q u e n e s s o f m i n i m a l p r o j e c t i o n f r o m 1 1 o n t o i t s s u b s p a c e g _ 1 ( o . ) i n t h e c a s e t h a t t h e r e i s a p e r m u t a t i o n £ p ( n ) ) o f p o s i t i v e i n t e g e r s and t h e s e q u e n c e ( £ n ) £ i = +1 o r -1 s u c h t h a t t h e l i n e a r f u n c t i o n a l f = ( £ g , ^)fe(l.. ) * s a t i s f i e s t h e c o n d i -v n p (n ) 1 t i o n ( 1 ) . As f a r as I kn ow , t h e r e m a i n i n g c a s e i s u n e x p l o r e d .) N o t e T h e p r o j e c t i o n ( = an i d e m p o t e n t b o u n d e d l i n e a r o p e r a t o r ) Pq f r o m a Banach s p a c e B o n t o i t s s u b s p a c e D i s m i n i m a l i f )|P0 H ^ l\P || f o r e v e r y p r o j e c t i o n P : B —} D. REFERENCES [ 1 ] B l a t t e r J . , Cheney E.W . , M i n i m a l P r o j e c t i o n s on H y p e r p l a n e s i n S e q u e n c e s S p a c e s , A n n a l i d i M a t . p u r a e d a p p l . , v . 1 0 1 , (1974 ),
