On equivalent strictly G-convex renormings of Banach spaces

Transcription

1 Cet. Eur. J. Math. 8(5) DOI: /s Cetral Europea Joural of Mathematics O equivalet strictly G-covex reormigs of Baach spaces Research Article Nataliia V. Boyko Departmet of Mechaics ad Mathematics, Kharkov Natioal Uiversity, Kharkov, Ukraie Received 2 Jauary 200; accepted 24 Jue 200 Abstract: MSC: We study strictly G-covex reormigs ad extesios of strictly G-covex orms o Baach spaces. We prove that l ω (Γ) space caot be strictly G-covex reormed give Γ is ucoutable ad G is bouded ad separable. 46B20 Keywords: Strict covexity Complex uiform covexity Strict G-covexity Versita Sp. z o.o.. Itroductio I this paper X stads for a Baach space (real or complex) ad L(X) for the algebra of bouded operators o X. By l (Γ) we deote the sup-ormed Baach space of all scalar-valued bouded fuctios o the set Γ. l ω (Γ) is the subspace of l (Γ) formed by elemets with coutable supports. For the basic defiitios ad stadard facts from Baach space geometry we refer to the Diestel s book [4], ad for Fuctioal Aalysis termiology we refer to [3] ad [5]. The cocept of strict G-covexity is a atural geeralizatio of classical cocepts of strict covexity (rotudity) ad complex strict covexity (rotudity). The idea of this geeralizatio cosists of substitutig the choice of sig (or of modulus-oe scalar) i the correspodig defiitios by the choice of operator from a give subset G L(X). Strict G-covexity ad the related cocept of uiform G-covexity were itroduced i [] ad were studied also i [2]. Let G L(X). X is said to be strictly G-covex if sup T G x + εt y > for all x, y S X ad ε > 0. For G = {I, I} ad G = {I, I, ii, ii} oe gets the classical cocepts of strict covexity ad complex strict covexity, respectively. Sice i these cases we deal with fiite groups of isometries which are symmetric about 0, fiite groups G with T = 0 () T G maletska.ata@gmail.com 87

2 O equivalet strictly G-covex reormigs of Baach spaces are of the major iterest for us. Nevertheless we cosider more geeral classes of G as well. Note that by imposig the coditio () we esure that at least sup T G x + εt y for all x, y S X ad ε > 0. So the strict G-covexity just meas that the last iequality is strict. I [2, Lemma 2.] oe ca fid more motivatio to impose (). There are a umber of well kow results about strictly covex spaces, i.e. about strictly G-covex spaces for G = {I, I}. The basic questio for us is what properties of G are resposible for these results. I our paper we study two theorems of this kid. Day s theorem [4] states that for ucoutable Γ, l (Γ) does ot have a equivalet strictly covex orm. We show that separability of G eables to prove the same result i the geeral case (Theorem 4.4), ad that there is a group G of cotiuum cardiality for which this result is false. Aother result is the Wee-Kee Tag extesio theorem [6]. We check that his costructio of a strictly covex orm extesio is applicable to arbitrary fiite groups G with T G T = 0 (Theorem 3.2). Recall that i our previous papers [] ad [2], we developed aalogous program for uiform G-covexity. Namely, for series i Baach spaces, we studied the coectio betwee G-covergece ad ucoditioal covergece i ordiary sese, as well as obtaied the aalogues of the theory of cotype ad the famous M. Kadets s theorem about ucoditioally coverget series i uiformly covex space. We also studied the iheritace of the uiform G-covexity of the space X by the space of X-valued fuctios L p (µ, X), p [, ) i the (most iterestig for us) case whe G is a regular fiite group. We showed that for a fixed p [, ) the uiform G-covexity of L p (µ, X) for all measure spaces (Ω, Σ, µ) is equivalet to a stroger property of X, i.e. to the uiform G-covexity i terms of p-average. We demostrated that for p (, ) the last property is equivalet to uiform G-covexity of X, but for p = this equivalece is o loger true. 2. Geeral remarks Theorem 2.. Let G be a fiite group, ad let X be a strictly G-covex space. The oe ca fid a equivalet orm o X, for which X edowed with this orm is strictly G-covex ad the group G cosists of isometries. Proof. Let us defie a ew orm by x := max T G T x. For the orm defied i such a way the followig iequalities hold: x x, because G is a group ad therefore it cotais the idetity operator; ad x max T G T x, the maximum of operator orms is fiite because the group G is fiite. These two iequalities imply the equivalece of the orms ad. For every T G T = sup x S U G (X, ) max UT x = sup x S (X, ) max Rx =. R G Sice G is a group this meas also that T = ad this implies that T is a isometry. Let us prove ow that the space (X, ) is strictly G-covex. For every positive real t ad for two arbitrary elemets x, y S (X, ) we have sup T G x + tt y = sup sup Ux + tut y = sup sup Ux + tut y T G U G U G T G ( = sup Ux sup Ux U G T G Ux + t y ( ) ) y Ux T y > sup Ux =. U G Note that i the last iequality we used strogly the fiiteess of G otherwise there would be o reaso to say that this iequality is strict. Theorem 2.2. Let G = {T,... T } be a fiite group, T G T = 0, X be a strictly covex space. The X is strictly G-covex. 872

3 N.V. Boyko Proof. Suppose that X is ot strictly G-covex, the there exist elemets x, y S X ad ε > 0 such that for all T G the followig iequality holds: x + εt y. The by covexity of the uit ball x + tt y for all t (0, ε). Sice X is strictly covex the x tt y > for all t (0, ε). So This is a cotradictio. < x ε T y = x + ε i=2 T i y x + εt i y. It is kow that for the space l oe ca fid a equivalet strictly covex orm. The the last theorem implies the followig corollary, which supplemets aturally our cosideratios from Sectio 3 below: i=2 Corollary 2.3. For every fiite group G L(l ) with T G T = 0 there is a equivalet strictly G-covex orm o l. Note also that strict G-covexity of a space does ot imply either ordiary strict covexity or complex strict covexity. Oe ca easily see this if oe cosiders X = l () (the space of vectors x = (x,..., x ) with x := max k x k ), ad G beig the (fiite ad symmetric) group of all isometries of X. 3. The extesio theorem Lemma 3.. Let p : B R + be a covex fuctio defied o a covex set B, ad for some collectio {x k } iequality ( ) p 2 (x k ) p 2 x k 0 B suppose the holds true. The p(x i ) = p(x j ) = p ( x k) for every i ad j. Proof. By covexity p ( x ) k p(x k). The 0 p 2 (x k ) p 2 ( ) ( x k ) p 2 (x k ) 2 p(x 2 k ) = 2 i<j (p(x i ) p(x j )) 2. Sice all the summads of the last sum are o-egative ad their sum is bouded by 0, the each summad is equal to 0. Recall that the Mikowski fuctioal of a covex absorbig rouded subset B of a liear space ([3, p. 06] or [5, sectio 5.4.2]) is φ B (x) = if {t > 0 : x tb}. Theorem 3.2. Let G = {T,... T } L(X) be a fiite group, i= T i = 0, ad suppose X has a strictly G-covex orm. The every equivalet strictly G-covex orm, defied o a subspace Y ivariat with respect to G, ca be exteded to a equivalet strictly G-covex orm, which is defied o the etire space X. 873

4 O equivalet strictly G-covex reormigs of Baach spaces Proof. Without loss of geerality we may assume that 2 r o Y for some r > 0. Let us exted the orm from Y to the etire space X i the stadard way, takig the closed covex hull of B (Y, ) B (X,r ) as the uit ball of the exteded orm. Further we will use for this extesio as well. The iequality 2 r is valid o the whole X after this extesio. Deote as dist(x, Y ) the distace from x to Y i the metric. Let us defie the fuctio p( ) as p 2 (x) = q (x) + q 2 (x) + q 3 (x), where q (x) = x 2, q 2 (x) = dist(x, Y ) 2 e x 2, ad q 3 (x) = dist(x, Y ) 2. Note that the fuctios q ad q 3 are covex o the whole X, ad q 2 is covex o B (X, ). I order to prove the latter it is eough to see that the Hessia of the fuctio f(x, y) = y 2 e x2, where (x, y) R 2 is positively defied o (0, 2 ) (0, 2 ). Therefore, p 2 is a covex fuctio o B (X, ). Defie B = {x X : p(x) }. Sice B B (X, ), we have the boudedess of B ad by covexity of p 2 we have the covexity of B as well. Moreover, for all x B (X, ) we have p 2 (x) 2 + e. By covexity of p 2 we have p 2 (tx) tp 2 (x) for x B (X, ), t (0, ). This meas that B B 2+e (X, ). The the Mikowski fuctioal of B defies a equivalet orm p ( ) o X. Let X be ot strictly G-covex i the orm p ( ). The there exist elemets x, y S (X,p ) B (X, ) ad t > 0 such that for all operators T k from the group G the iequality p (x + tt k y) holds. The, p 2 (x + tt k y) p 2 (x) 0. Recall that p 2 = q (x) + q 2 (x) + q 3 (x) is a sum of covex fuctios, ad by covexity each summad satisfies the iverse iequality: q j (x + tt k y) q j (x) 0, j =, 2, 3 (at this poit we use the coditio T k = 0). So i fact all these iequalities are equatios: q j (x + tt k y) q j (x) = 0, j =, 2, 3. I particular dist(x + tt k y, Y ) 2 dist(x, Y ) 2 = 0. So accordig to the Lemma 3., for all k {, 2,.., } dist(x + tt k y, Y ) = dist(x, Y ). (2) Cosider two cases: whe x belogs to Y, ad whe it is out of this subspace. I the first case (2) implies that x + tt k y also belog to Y for all the operators T k G. Therefore, p 2 (x + tt k y) = x + tt k y 2, ad the assumptio that p (x + tt y) for all k cotradicts strict G-covexity of the orm defied o Y. I the secod case (2) together with q 2(x + tt k y) q 2 (x) = 0 implies that e x+tt k y 2 e x 2 = 0. Thus, j= ( j! ) x + tt k y 2j x 2j =

5 N.V. Boyko By the same covexity argumet as above this meas that x + tt k y 2i x 2i = 0 for all i N. But by the Lemma 3. already i the case of i =, x + tt k y 2 x 2 implies that x + tt k y = x for all k, which cotradicts strict G-covexity of the orm Thereby both cases lead to a cotradictio ad this proves that our assumptio was icorrect. Thus, the orm p is strictly G-covex o the etire space X. If the group G equals {I, I, ii, ii}, the G-covexity is the same as complex covexity. Therefore we get the followig corollary: Corollary 3.3. Let X have a complex strictly covex orm. The every equivalet complex strictly covex orm, defied o a subspace Y ca be exteded to a equivalet complex strictly covex orm defied o the etire space X. 4. Reormigs of l (Γ) It is atural to study the existece of Baach spaces which caot be strictly G-covex reormed for ay fiite group of isometries G. As it is show below, the space l (Γ) where Γ is ucoutable is a example of such a space. From ow o Γ is a ucoutable set. For every A B we cosider l ω (B \ A) as the subspace of l ω (B) cosistig of A-supported elemets of l ω (B). Lemma 4.. For every fuctioal x l ω (Γ) there exists a at most coutable set A Γ such that x (y) = 0 for all y l ω (Γ \ A). Proof. For every A Γ deote u(x, A) = sup x Slω(Γ\A) x (x). Let us cosider two cases: if A ω u(x, A) = 0 ad if A ω u(x, A) 0. I the first case for every there exists a at most coutable set A with u(x, A ) <. This way the set A = = A is at most coutable. A A, u(x, A) u(x, A ), hece u(x, A) = 0. So the lemma is proved for this case. I the secod case, let us fix a positive α < if A ω u(x, A). There exists a elemet x S lω(γ): x (x ) α, A = supp x. Oe ca also fid a elemet x 2 S lω(γ\a ): x (x 2 ) α, A 2 = supp x 2 supp x etc. We obtai a disjoit sequece (x k of orm elemets with x (x k ) α. Deote λ k = sig x (x k ). The λ kx k = ad x x ( λ ) kx k α + as. This is a cotradictio. So the first case is the oly possible case. Lemma 4.2. Let G be a coutable bouded family cosistig of operators T k : l ω (Γ) l ω (Γ), k N ad let A Γ be at most coutable. The oe ca fid a at most coutable set B A such that for every elemet x l ω (Γ \ B) ad every operator T from G the elemet T (x) belogs to l ω (Γ \ A). Proof. Deote by e γ l ω (Γ) the correspodig coordiate fuctioal: e γ(x) = x(γ). Cosider fuctioals xk,a (x) := e a(t k (x)), a A, k N. Applyig the previous lemma for all a A, k N we ca fid a at most coutable set A k,a Γ such that xk,a l ω(γ\a k,a ) = 0. The B k := a A A k,a is coutable ad has the followig property: for every x l ω (Γ \ B k ) the value of T k (x) belogs to l ω (Γ \ A). The set B that we eed ca be defied as B := B k. 875

6 O equivalet strictly G-covex reormigs of Baach spaces Lemma 4.3. Let p be a arbitrary equivalet orm o l ω (Γ). The there exists a o-zero elemet x l ω (Γ) ad exists a coutable set A Γ such that for every y l ω (Γ \ A), y, the equality p(x + y) = p(x) holds. Proof. For every elemet x S lω(γ) defie F x = {y S lω(γ) : y supp(x) = x supp(x) }. Without loss of geerality we ca assume that p( ). Defie M x = sup {p(y) : y F x }, m x = if {p(y) : y F x }. It is obvious that 2x F x = F x, hece p(2x y) M x, so 2p(x) M x + p(y). Moreover for every ε > 0 there exists a y F x, such that p(y) m x + ε. This way we obtai that for every ε > 0 the iequality 2p(x) m x + M x + ε holds. Thus the followig iequality holds as well: 2p(x) m x + M x. (3) Defie K = sup {p(y) : y S lω(γ)}. It is obvious that K ad let us choose a x S lω(γ) such that (3K +)/4 p(x ). The iequality (3) implies that (3K + )/2 m x + M x, the M x m x (K )/2. Choose a x 2 F x such that (3M x + p(x ))/4 p(x 2 ). Thus, (M x + p(x ))/2 m x2. It is obvious that M x2 M x, because x 2 F x ad F x2 F x. Hece M x2 m x2 (M x p(x ))/2 (M x m x2 )/2 K. 4 Iductively let us choose x F x ad we have M x m x (K )/2, M xk+ M xk, m xk+ m xk. Therefore there exists lim k M xk = lim k m xk = µ. It is obvious that p(y) = µ for every elemet y F x. Choose a elemet x l ω (Γ), such that x(ν) = x (ν) for all N ad ν supp (x ), ad x(ν) = 0 outside A := N supp (x ). For this x the correspodig set F x cosists of elemets w with p(w) = p(x). Sice every elemet of the form x + y: y l ω (Γ \ A), y, belogs to F x, we are doe. Theorem 4.4. If Γ is ucoutable ad G is a separable bouded operator family, the l ω (Γ) caot be equivaletly reormed i strictly G-covex way. Proof. Let p(x) be a arbitrary equivalet orm o l ω (Γ). By Lemma 4.3 there exist a x l ω (Γ)\{0} ad a coutable set A Γ, such that for every y l ω (Γ\A), y, the equality p(x +y) = p(x) holds. Sice the group G is separable, the it has a coutable dese subset G. Accordig to Lemma 4.2, for a give set A we ca fid a set B A, such that for all y from B lω(γ\b) ad for all T G the elemets T y belog to l ω (Γ \ A). Sice the group G is bouded there exists a t > 0, such that tt y for all y ad all T G. The, for all the elemets y B lω(γ\b), the followig holds: sup T G p(x + tt y) = p(x). By desity of G i G ad cotiuity of orm the same is true for G: sup T G p(x + tt y) = p(x). This meas that (l ω (Γ), p) is ot strictly G-covex. Let us remark that the coditios of boudess ad separability of G caot be removed. At first, i every space X if oe takes G {I, 2I, 3I,...} the X is strictly G-covex. By this reaso oly bouded collectios of operators are cosidered. Further, if oe cosiders the group G o l ω (Γ) geerated by I, I ad operators of pairwise rearragemets of coordiates, the G has the same cardiality as Γ ad so it ca, for example, be equal to the first ucoutable cardial. O the other had l ω (Γ) i its origial orm is strictly G-covex (ad eve uiformly G-covex) with respect to this group of isometries. Ackowledgemets The author expresses deep gratitude to her thesis advisor Vladimir Kadets for multiple valuable suggestios, ad for the aoymous referee, whose advises sigificatly improved the expositio. 876

7 N.V. Boyko Refereces [] Boyko N., O arragemet of operators coefficiets of series member, Vis. Khark. Uiv., Ser. Mat. Prykl. Mat. Mekh., 2008, 826(58), 97 20, (i Russia) [2] Boyko N., Kadets V., Uiform G-covexity for vector-valued L p spaces, Serdica Math. J., 2009, 35, 4 [3] Coway J.B., A Course i Fuctioal Aalysis, 2d ed., Graduate Texts i Mathematics, 96, Spriger, New York, 990 [4] Diestel J., Geometry of Baach Spaces Selected Topics, Lecture Notes i Mathematics, 485, Spriger, New York, 975 [5] Kadets V.M., A Course i Fuctioal Aalysis. Textbook for studets of mechaics ad mathematics, Kharkov State Uiversity, Kharkov, 2006, (i Russia) [6] Tag W.-K., O the extesio of rotud orms, Mauscripta Math., 996, 9(),