ON THE STRUCTURE OF THE SPREADING MODELS OF A BANACH SPACE

Transcription

1 ON THE STRUCTURE OF THE SPREADING MODELS OF A BANACH SPACE G. ANDROULAKIS, E. ODELL, TH. SCHLUMPRECHT, N. TOMCZAK-JAEGERMANN Abstract We study some questos cocerg the structure of the set of spreadg models of a separable fte-dmesoal Baach space X. I partcular we gve a example of a reflexve X so that all spreadg models of X cota l but oe of them s somorphc to l. We also prove that for ay coutable set C of spreadg models geerated by weakly ull sequeces there s a spreadg model geerated by a weakly ull sequece whch domates each elemet of C. I certa cases ths esures that X admts, for each α < ω, a spreadg model ( x (α) ) such that f α < β the ( x (α) ) s domated by (ad ot equvalet to) ( x (β) ). Some applcatos of these deas are used to gve suffcet codtos o a Baach space for the exstece of a subspace ad a operator defed o the subspace, whch s ot a compact perturbato of a multple of the cluso map.. Itroducto sec It s kow that for every semormalzed basc sequece (y ) a Baach space ad for every ε ց 0 there exsts a subsequece (x ) ad a semormalzed basc sequece ( x ) such that: For all N, (a ) [, ] ad k <... < k eq0 () a x k a x < ε. The sequece ( x ) s called the spreadg model of (x ) ad t s a suppresso- ucodtoal basc sequece f (y ) s weakly ull (see [4] BS ad [5]; BS2 see also [3](I.3.Proposto BL 2) ad [20] O for more about spreadg models). Ths coucto wth Rosethal s l theorem [25], R yelds that every separable fte dmesoal Baach space X admts a suppresso - ucodtoal spreadg model ( x ). I fact oe ca always fd a -ucodtoal spreadg model [26]. R2 It s atural to ask f oe ca always say more. What types of spreadg models must always exst? Sometmes we refer to the closed lear spa of ( x ), as the spreadg model of (x ). By James well kow theorem [2] J every such X thus admts a spreadg model X whch s ether reflexve or cotas a somorph of c 0 or l. It was oce speculated that for all such X some spreadg model ( x ) must be equvalet to the ut vector bass of c 0 or l p for some p < but ths was proved to be false [2]. OS A replacemet coecture was brought to our atteto by V.D. Mlma: must every separable space X admt a spreadg model whch s ether somorphc to c 0 or l or s reflexve? I secto 2 we show ths to be false by costructg a space X so that for all spreadg models X of X, X cotas l but X s ever somorphc to l. The example borrows some of the tuto behd the example of [2]. OS That space had the property that amogst the l p ad c 0 spaces Ths research was supported by NSF, NSERC ad the Pacfc Isttute for the Mathematcal Sceces. I addto, the fourth author holds the Caada Research Char Mathematcs.

2 oly l could be block ftely represetable ay spreadg model ( x ) yet o spreadg model could cota l. The motvato behd our example comes from the Schreerzed verso S(d w, ) of the Loretz space d w,. Let = w > w 2 >... wth w 0 ad = w =. The d w, s the sequece space whose orm s gve by x = w x where x s the sequece (x ) ad (x ) s the decreasg rearragemet of ( x ). Oe could the defe the sequece space S(d w, ) as the completo of c 00 (the lear spa of ftely supported sequeces of reals) uder x = sup N, k <k 2 <...<k w x k. I ths case the ut vector bass (e ) has a spreadg model, amely the ut vector bass of d w,, whch s ot a l bass but whose spa s heredtarly l. S(d w, ) s heredtarly c 0 so t does ot solve Mlma s questo. I order to avod c 0 oe may also defe the Tsrelsozed verso T(d w, ) of d w,. T(d w, ) s the completo of c 00 uder the mplct equato ( ) x = max x, sup w E x where the supremum s take over all tegers, ad all admssble sets (E ).e. E <... < E (ths meas m E maxe < m E 2...). E x s the restrcto of x to the set E. It may well be that T(d w, ) has the propertes we desre but we were uable to show ths. Thus we were forced to layer the orm a certa sese (see secto 2 below). I Secto 3 we cosder a wder cotext SP ω (X), the partally ordered set of all spreadg models ( x ) geerated by weakly ull sequeces X. The partal order s defed by domato: we wrte ( x ) (ỹ ) f for some C < we have C a x a ỹ for all scalars (a ). We detfy ( x ) ad (ỹ ) SP ω (X) f ( x ) (ỹ ) ( x ). We prove ( Proposto 3.2) malemma that f C SP ω (X) s coutable the there exsts ( x ) SP ω (X) whch domates all members of C. Ths eables us to prove that certa cases oe ca produce a ucoutable cha {( x (α) ) } α<ω wth ( x (α) ) < ( x (β) ) f α < β < ω. The example of the prevous secto ad the above yeld a soluto to a uformty questo rased by H. Rosethal. The questo (ad a dual verso) are as follows: Let a separable Baach space Z have the property that for all spreadg models ( x ) of ormalzed basc sequeces lm x / = 0 ( ) respectvely, lm x =. Does there exst (λ ) wth lm λ / = 0 (respectvely lm λ = ) such that for all spreadg models ( x ) of ormalzed basc sequeces Z ( ) lm x /λ = 0 respectvely, lm x /λ =? 2

3 We gve egatve aswers to these questos. The example that solves the frst questo s the space X of secto 2. Moreover every subspace of X fals to admt such a sequece (λ ). We do ot kow of a heredtary soluto to the secod questo. I secto 5 we cosder the problem: f SP ω (X) =,.e., f X has a uque spreadg model up to equvalece, must ths spreadg model be equvalet to the ut vector bass c 0 or l p for some p <? The questo was asked of us by S. A. Argyros. It s easy to see that the aswer s postve f the spreadg models are uformly somorphc. We show that the aswer s postve f belogs to the Krve set of some spreadg model. def. Defto.. Let (x ) be a -spreadg basc sequece (see ( 2)). Cspreadg The Krve set of (x ) s the set of p s ( p ) wth the followg property: For all ε > 0 ad N there exsts m N ad (λ k ) m k= R, such that for all (a ) R, + ε (a ) p a y ( + ε) (a ) p where y = m k= λ kx ( )m+k for =,...,, ad p deotes the orm of the space l p. The proof of Krve s theorem [4] K as modfed by H. Lemberg [5]) L (see also [9], G Remark II.5.4 ad [8]), MS shows that for every -spreadg basc sequece (x ) the Krve set of (x ) s o-empty. It s mportat to ote that our defto of a Krve p requres ot merely that l p be block ftely represetable [x : N] but each l p ut vector bass s obtaable by meas of a detcally dstrbuted block bass. A mmedate cosequece of the fact that the Krve set of a spreadg model s oempty s the followg: rem. Remark.2. Assume that (x ) s a semormalzed basc sequece a Baach space X whch has a spreadg model ( x ). We ca assume that for some decreasg to zero sequece (ε ) ( ) eq0 s satsfed. The there s a p [, ] such that for all ad all ε > 0 there exsts a fte sequece (λ ) m R so that ay block (y ) of (x ) of the form m y = λ x (,), wth (, ) < (, 2) <... < (, m) < (2, ) <...(2, m) < (3, )... = has a spreadg model (ỹ ) whch s sometrc to the sequece ( m = λ x ( )m+ ) N ad has the property that ts frst elemets are ( + ε)-equvalet to the ut vector bass of l p. For 0 N large eough (or passg to a approprate subsequece of (x )) we also observe that (y k ) = s ( + 2ε)-equvalet to the l p ut bass wheever 0 < k <... k. I Secto 6 sec5 we gve suffcet codtos o a Baach space X for the exstece of a subspace Y of X ad a operator T : Y X whch s ot a compact perturbato of the cluso map. W.T. Gowers [9] G proved that there exsts a subspace Y of the Gowers-Maurey space GM (costructed [0]) GM ad there exsts a operator T : Y GM whch s ot a compact perturbato of the cluso map. Here we exted the work of Gowers to a more geeral settg. For example suppose that X admts a spreadg model ( x ) whch s ot equvalet to the ut vector bass l but such that s the Krve set of ( x ). The (Theorem 6.) Ma3 there exsts a subspace W of X ad a bouded operator T : W W such that p(t) s ot a compact perturbato of the detty, for ay polyomal p. 3

4 Our termology s stadard as may be foud LT [6]. All our Baach spaces wll be cosdered spaces over the real feld R. If A X, where X s a Baach space, the spa(a) s the lear spa of A ad [A] = spa(a) s the closed lear spa of A. If S s a set, c 00 (S) deotes the vector space of ftely supported real valued fuctos o S. If S = N we wrte c 00 = c 00 (N). S X s the ut sphere of X ad B X s the ut ball of X. A basc sequece (x ) s block ftely represeted (y ) f for all ε > 0 ad N there exsts a block bass (z ) of (y ) satsfyg ( + ε) a x a z ( + ε) a x for all (a ) R. l p s block ftely represeted (y ) f the ut vector bass of l p s block ftely represeted (y ). Let (x ) be a basc sequece ad C. (x ) s called C-spreadg f for all (a ) c 00 ad all choces of < 2 <... N,, Cspreadg (2) a x a x C a x C sec2 ad (x ) s called C-suppresso ucodtoal f for all (a ) c 00 ad A N.. a x C a x A We say that (x ) s C-subsymmetrc f t s C-spreadg ad C-suppresso ucodtoal. Here we slghtly devate from the otos [6], LT where C-subsymmetrc s defed to be C-spreadg ad C-ucodtoal (wth respect to chages of sgs). We say that (x ) s spreadg, ucodtoal, or subsymmetrc, f for some C, (x ), s C-spreadg, C- ucodtoal, or C-subsymmetrc, respectvely. We thak the referee for hs or her pastakg effort whch saved us from some embarrassg gltches. 2. Spreadg models cotag l whch are ot l Let us start wth a observato whch wll be used several tmes through out the paper. prop2.0 Proposto 2.. Assume that (f ) s a ormalzed subsymmetrc basc sequece. The followg codtos are equvalet. a) (f ) s equvalet to the ut vector bass of l. b) There s a r > 0 so that r, for all N. a (ρ) f c) There s a C > 0 such that for all ρ > 0 there exsts a (a (ρ) f = ad a (ρ) C. ) c 00 [ ρ, ρ] N, so that Proof. Clearly (a) (b) (c). To prove the coverse, we frst assume w.l.o.g. that (f ) s -subsymmetrc. Let f, N, be the coordate fuctoals. Sce (f ) s also a - subsymmetrc basc sequece, we oly eed to show that the partal sums ( f ) are bouded the dual orm. 4

5 Let ρ > 0 be arbtrary but fxed. Choose x ρ = b (ρ) f Sspa(f : N) so that x ρ ( a (ρ) f ) = b (ρ) a (ρ) =. By ucodtoalty we ca assume that sg(b (ρ) ) = sg(a (ρ) ) = +, for =, 2,.. ad deduce that ρ { : b (ρ) 2C } N,b (ρ) >/(2C) a (ρ) b (ρ) = N,b (ρ) /(2C) Ths mples, aga by the fact that (f ) s -subsymmetrc that 2ρ 2C f ad fshes the proof, f we let ρ 0. N,b (ρ) /(2C) b (ρ) f x ρ =, a (ρ) b (ρ) 2. thm2. Theorem 2.2. There exsts a reflexve Baach space X wth a ucodtoal bass such that the spreadg model of ay ormalzed basc sequece X s ot somorphc to c 0 or l ad s ot reflexve. For x = (x ) c 00 we wrte supp x = { : x 0}. For x, y c 00 ad a teger k we say that x < y f max supp x < m supp y, ad we wrte k < x f k < m supp x. (e ) deotes the ut vector bass of c 00. I order to prove Theorem 2.2 thm2. we wll costruct a space X whch has certa propertes as stated the followg result, whch wll easly mply Theorem 2.2. thm2. thm2.2 Theorem 2.3. There s a space X wth the followg propertes: a) X has a ormalzed -ucodtoal bass (e ). b) For ay ormalzed block bass of (e ) havg a spreadg model ( x ) we have that ( x ) s ot equvalet to the ut vector bass of l. c) For ay ormalzed block bass of (e ) havg a spreadg model ( x ) we have that l embeds to spa({ x : N}). Proof of Theorem 2.2. thm2. Let X be chose as Theorem 2.3. thm2.2 Sce X has a ucodtoal bass ad does ot cota a subspace somorphc to l or c 0 (otherwse a block bass of (e ) would be equvalet to ether the ut vector bass of l or c 0, both of whch are excluded by (b) ad (c)), X must be reflexve. Sce X s reflexve every ormalzed basc sequece X has a subsequece whch s equvalet to a block bass of (e ). Therefore (b) ad (c), ad the fact that l has a uque subsymmetrc bass, mply that all the spreadg models of ormalzed basc sequeces X are ether reflexve or somorphc to c 0 or l. Costructo of the space X: Frst we choose a creasg sequece of tegers ( ) such that k eq (3) ( k ) /p 3 for all p >. k 5

6 I order to choose a sequece ( ) satsfyg ( eq 3), frst choose a sequece (p k ) k wth p k ց ad the ductvely o k N pck ( k ) to satsfy ( k ) /p k k 3 > k for all k N. Now we choose a orm o c 00 to satsfy the followg Tsrelso type equato (see OS2 [23]): x = x k sup k N k E () <E() 2 <...<E(),for k 3 = E () x. Note that we do ot requre that E (s) E (t) = f s t. Heceforth ths secto X wll deote the completo of c 00 uder ths orm. It s easy to see that the ut vector bass (e ) s a ormalzed -ucodtoal bass for X. It wll be useful to troduce the sequece of equvalet orms, for N, as follows: Note that we have x = sup E <E 2 <...<E x = x sup k N k = E x. 3 [k, )x. Proof of Theorem thm a) s mmedate. b) We eed the followg auxlary results. We postpoe the proofs. lem2.3 Lemma 2.4. For ay ormalzed block bass (y ) of (e ) ad for ay ε > 0 there exsts a subsequece (x ) ad 0 N such that for ay N N ad tegers k,,..., N wth 0 k < 2 <... < N we have that eq2 (4) k = 0 3 [k, ) ( N ) N x s < ε. lem2.4 Lemma 2.5. Let (y ) be a ormalzed block bass of (e ) X whch has a spreadg model (ỹ ) ad suppose that N N satsfes eq3 (5).99 2N (ỹ ỹ 2N ). The there exsts k N ad a subsequece (x ) of (y ) such that for all < 2 <... < N, ( ) k [k, N eq4 (6).96 < ) x 3 s. N For the proof of b) assume to the cotrary that there exsts a ormalzed block bass (y ) of (e ) whose spreadg model (ỹ ) s equvalet to the ut vector bass of l. Wthout loss of geeralty [3](Proposto BL 4 Chapter II Secto 2), we ca assume that ( 5) eq3 s vald for 6 s= s=

7 all N N. For ε =.0 choose 0 N ad a subsequece of (y ) whch satsfes the cocluso of Lemma 2.4. lem2.3 Choose N N wth 2 0 <.0. N Sce ( 5) eq3 s vald, by Lemma 2.5 lem2.4 there exsts k N ad a further subsequece (x ) whch satsfes ( 6). eq4 Now let < 2 <... < N wth k ad let x = (/N) N s= x s. We wll frst estmate for N the value of x. Choose E < E 2 <... < E so that x = E (x). Sce (e ) s -ucodtoal we ca assume that the E s are tervals N, that m E =, ad that maxe = m E +, for =,...,. For l =, 2,..., put I l = {s N : supp(x s ) E l } ad I 0 = {, 2,..., N} \ l= I l ad ote that I 0 = { s N : l, l 2, l l 2, supp(x s ) E lt, t =, 2 }, ad that l I l N. Moreover ote that each E l ca oly have a o empty tersecto wth the support of at most two x s s, s I 0. Therefore we deduce [ ] eq3a (7) x = E l (x) ( ) x s + E l x s + 2 N N. l= l= s I l s I 0 By Lemma 2.5 lem2.4 we have (the secod term below o the rght dsappears f k < 0 ) 0 k.96 < 3 [k, )x + 3 [k, )x = x +.0 (by ( eq2 4) sce k ) N 3 N +.0 < =.52 (by ( eq3a 7)) whch s a cotradcto. c) Here we eed the followg result whose proof s aga postpoed: lem2.5 Lemma 2.6. Let (z ) be a ormalzed block bass of (e ) wth spreadg model ( z ). The for every K N there exsts a K 2 > K ad (w ), a detcally dstrbuted block bass of (z ), whch has a spreadg model ( w ) (whch s a block bass of ( z )) such that for all l N:.98 w l ad eq5 (8) K 2 =K + 3 [K 2, )w l >.4. Let (z ) be a ormalzed block bass of (e ) havg a spreadg model ( z ). By passg to a subsequece f ecessary we ca assume that ( ) eq0 s satsfed for some sequece (ε ) whch coverges to 0. By applyg Lemma 2.6 lem2.5 repeatedly, there exsts a creasg sequece of tegers (K ), (K = 0), ad for every N there exsts a detcally dstrbuted block 7

8 bass (w () ) of (z ) havg spreadg model ( w () ), whch s also a block bass of ( z ), such that for all, l N,.98 w () l ad eq6 (9) K + =K + 3 [K +, )w () l >.4 Choose a sequece (m ) of tegers such that ( w m () ) s a block sequece of ( z ). We clam that ( w m () ) s equvalet to the ut vector bass of l. We show that for N N ad (a ) N R, N a w m () N >.4 a. Let < 2 <... be such that w () < w (2) 2 w () 2N+ <.... The, sce (z ) satsfes ( eq0 <... < w (N) N < w () N+ < w (2) N+2 <... < w (N) ), t follows that N a w m () N = lml = a w () (l )N+. If we choose l such that N = w() (l )N+ s supported o [K N+, ) the 2N < eq7 (0) N = KN+ a w () [KN+ (l )N+, ) 3 N a = K + =K + N = a w () (l )N+ 3 [K +, )w () (l )N+ >.4 N = a (by ( eq6 9)). Proof of Lemma 2.4. lem2.3 Sce for all ad we have y, by a smple compactess ad dagoalzato argumet there exsts a subsequece (x ) of (y ) such that eq8 () x x for all. eq9 (2) Now we clam that 3 x 3 2. Ideed, otherwse there exsts k N such that eq0 (3) k 3 x >

9 Choose k such that x s supported o [k, ). The k x 3 [k, )x = k k 3 x (sce x s supported o [k, )) 3 ( x ) (by ( eq8 ) sce k) > 3 k 2 > (by (eq0 3)) 3 whch s a cotradcto. Thus ( 2) eq9 s establshed. Now choose 0 N such that eq (4) 3 x + 3 < ε. = 0 = 0 Let k,,..., N N wth 0 k < 2 <... < N. We have k [k, N N k ) x 3 s 3 x s = 0 s= s= = 0 N k 3 ( x + ) (by ( ) eq8 sce k ) s= = 0 < Nε (by ( 4)). eq Proof of Lemma lem From ( eq3 5) there exsts a subsequece (z ) of (y ) such that eq2 (5).98 < 2N (z + z z N + z + z z N ) for all N < < 2 <... < N. Let K be the maxmum elemet the support of z N. Now for < 2 <... < N let u = (z z N )/N, v = (z z N )/N ad w = (u + v)/2. By the defto of the orm of X there exsts k N, whch depeds o,..., N, such that w = k 3 [k, )w. By ( 5) eq2 we have that.98 < w ad thus k K. By the tragle equalty we obta.98 < 2 k 2 u [k, )u + 2 k k 3 [k, )v 3 [k, )v k 3 [k, )v.

10 Thus eq3 (6).96 < k 3 [k, )v. Now by Ramsey s theorem [24] Ra (see also [20]) O there exsts a subsequece (x ) of (z ) N ad k K such that k (, 2,... N ) = k for all choces of < 2 <... < N, ad, thus ( 6) eq4 s vald for all < 2 <... < N. Proof of Lemma 2.6. lem2.5 Let us frst ote that ether l p, p >, or c 0 are ftely block represeted X. Ideed, f (x ) for =,..., k (for some k N) s a ormalzed block bass of (e ) whch s 2-equvalet to the frst k ut basc vectors of l p for some p >, the f supp x > k, t follows that 2( k ) /p = k k k 3 = 3 x x (by defto of the orm) whch cotradcts ( 3). eq Smlarly the case p = s excluded ad thus the coclusos of Remark.2 rem. hold oly for p =. Let (z ) be a ormalzed block sequece X havg a spreadg model ( z ), ad let K N. Choose N N such that 2 N K <.0. By Remark.2 rem. there exsts a detcally dstrbuted block bass (y ) of (z ) havg spreadg model (ỹ ) whch satsfes ( 5) eq3 ad (ỹ l ) s a block bass of ( z ). Thus by Lemma 2.5 lem2.4 there exsts K 2 N ad a subsequece (x ) of (y ) such that ( 6) eq4 s satsfed for k = K 2 ad for all < 2 <... < N. Let w l = N x N(l )+ for l N. N = Sce ( 5) eq3 s satsfed, by passg to a subsequece we ca assume that.98 w l for all l. Let ( w ) be the spreadg model of (w ). The for all l N, Thus ( w l ) s a block bass of ( z ) ad eq4 (7).96 < w l = N K 2 N ỹ N(l )+. = 3 [K 2, )w l. 0

11 eq5 (8) sec3 Note also that by wth the same argumet as the proof of ( 7), eq3a K K 3 [K 2, )w l 2 + N 3 N < =.5. (by ( eq3 6)) Now ( eq4 7) ad ( eq5 8) mmedately gve ( eq5 8). We recall the stadard 3. The set of spreadg models of X D3. Defto 3.. Let (x ) ad (y ) be basc sequeces ad C. We say that (x ) C- domates (y ), f C a x a y for all (a ) c 00. We say that (x ) domates (y ), deoted by (x ) (y ), f (x ) C-domates (y ) for some C. We wrte (x ) > (y ), f (x ) (y ) ad (y ) (x ). If B s a set of basc sequeces ad (z ) s a basc sequece, the we say that (z ) uformly domates B there exsts C such that (z ) C-domates every elemet of B. The set SP(X) of all spreadg models geerated by ormalzed basc sequeces X s partally ordered by domato, provded that we detfy equvalet spreadg models. SP ω (X) deotes the subset of those spreadg models geerated by weakly ull sequeces. Our frst result ths secto shows that every coutable subset of SP ω (X) admts a upper boud SP ω (X). malemma Proposto 3.2. Let (C ) (0, + ) be such that C < ad for N let (x () ) be a ormalzed weakly ull sequece some Baach space X havg spreadg model ( x () ). The there exsts a semormalzed weakly ull basc sequece (y ) X wth a spreadg model (ỹ ) havg the followg propertes. a) (ỹ ) C -domates ( x () ) for all N. b) If for o N, ( x () ) s equvalet to the ut vector bass of l, the (ỹ ) s ot equvalet to the ut vector bass of l. c) If (z ) s a basc sequece whch uformly domates ( x () ) for all N, the (z ) domates (ỹ ). I order to prove Proposto 3.2 malemma we frst eed to geeralze the fact that spreadg models of ormalzed weakly ull sequeces exst ad are suppresso -ucodtoal. Lemma 3.3 lem3.3b s actually a specal case of a more geeral stuato []. HO The results could also be phrased terms of coutably brachg trees of order m ad proved much lke the argumets [3]. KOS lem3.3b Lemma 3.3. Let, m N ad ε > 0. Let (x () ), (x (2) ),...,(x () ) be ormalzed weakly ull sequeces a Baach space X. The there exsts a subsequece L of N so that for all famles of tegers (k () ),m,=... < k () 2 <... < k m () (a () ) m,,= [, ] we have m ad (l() ),m,= L, wth k() < k (2) <...k () < k () 2 < <... < k () m ad l () < l (2) <... < l () < l () 2 <... < l () = a () x () l () m = a () x () k () ε. m, ad

12 Proof. Ths follows easly by Ramsey s theorem. Let (a () ) m,,= [, ]. Partto [0, m] to ftely may tervals of legth less tha ε/2. Partto the sequeces of legth m, k () < k (2) <...k () < k () 2 <... < k m () of N, accordg to whch terval m = a() x () belogs. Thus by Ramsey s theorem for some fte subsequece L of k () N these expressos belog to the same terval, f k () We repeat ths for a fte ε/4-et of [, ] m edowed wth the l m L for =,..., m ad =,...,. orm. ),...,(x () ) be ormalzed weakly ull sequeces lem3.3d Lemma 3.4. Let, m N, ε > 0 ad (x () a Baach space X. The there exsts a subsequece L of N so that for all tegers L, k () < k (2) <...k () < k () 2 <... < k () 2 <... < k m () form a suppresso ( + ε)-ucodtoal basc sequece. <... < k () m, the vectors (x () k () ) m,,= Proof. By passg to subsequeces, f ecessary, we may assume that the sequece (x () satsfes the cocluso of Lemma 3.3 lem3.3b for ε replaced by ε/2 ad L = N. Let δ = ε/(2m). ), =, We clam that for every 0 ad 0 N there exsts > 0 such that for every fuctoal f X of orm there exsts [ 0, ] wth f(x ( 0) ) < δ. Ideed, assume that such a > 0 dd ot exst. The we could fd for each > 0 a f S X such that f (x ( 0) ) δ for all { 0, 0 +,... }. Let f be a w -accumulato pot of the set {f : 0 }. It follows that f (x ( 0) ) δ for all 0, whch cotradcts the assumpto that (x ( 0) ) s weakly ull, ad proves the clam. Iteratg ths clam we ca pass to a fte subsequece L of N wth the followg property: For k () < k (2) <... < k () < k () 2 <...k m () L, F {k (), k(2),..., k(), k() 2,...,k() m }, ad f X of orm, there exst l () < l (2) <... < l () < l () 2 <... < l () m N wth l () = k () f k () F ad f(x () < δ f k () F. Let k () < k (2) <... < k () < {(,):k () F } k () l () k () 2 <... < k m () L, F {k (), k(2),...,k(), k() 2,...,k() m }, ad (a () m = a() x () =. There exsts f X of orm such that k () ( ) a () x () = f a (), ad choosg (l () ( m f {(,):k F } m = = a () a () x () k () x () l () x () l () ) + δm + ε 2 ( + ε 2 ) + ε 2 = + ε. ) m,,= [, ] wth ) as above, Proof of Proposto 3.2. malemma Usg Lemma 3.4, lem3.3d a dagoal argumet ad relabelg we ca assume that for all l ad all choces of l k () < k (2) <...k (l) < k () 2 <...k (l) l the vectors ), l are suppresso 2-ucodtoal. (x () k () Let m = 0 ad for N let m + = m +. Let (C ) (0, ) such that C <. By passg to the same subsequeces of (x () ), for each N, we ca assume addto 2

13 that the semormalzed sequece (y ), where y = 6C x () m + for all N, has a spreadg model (ỹ ). It s easy to check that (y ) s weakly ull sce each (x () ) s weakly ull for each N. Let 0, m N ad (a ) m = R. Let N such that max(m, 0) ad 8 m a = =+ C C 0 m = a x ( 0). I addto choose so that m 2 = a x ( 0) l m = a x ( 0) 2 m = a x ( 0) l ad m 2 = a y l m = a ỹ 2 m = a y l for all choces of l < l 2 <... < l m. We have from Lemma 3.4 lem3.3d ad our equaltes, f k < k 2 <...k m, m m B* (9) a ỹ a y k 2 A* (20) = = = k m a 6C x () = m = m = a 6C x () m k + m k + 2 a 6C 0 x ( 0) m k + 0 C m a 6C = 0 x ( 0) m k = =+ 0 m C m 0 = = a x ( 0) a x ( 0) a 6C = C m 0 = a x ( 0) Ths proves the part (a) of the proposto. I order to show the remag parts let m N ad (a ) m = R, ad frst ote that m m = a ỹ = lm... lm m = lm... lm m s= a s y s m s a s s= lm sup... lm sup m 6C 6C x () m s + m a s x () s= m s + = 6C m s=. a s x () s Part (c) ow follows from ( 20) A* ad the assumpto that (C ) s summable. I order to show part (b) assume that for ay N ( x () ) s ot equvalet to l ad let δ > 0. Frst choose 0 so that 6 = 0 + C < δ/2. The choose N large eough so that N N = x() δ/(2 0 ), 3.

14 for =,..., 0 (usg Proposto 2.), prop2.0 ad fally apply ( 20) A* for m = N, a =, to obta N N N = ỹ < δ. creas Remark 3.5. Usg a smlar argumet we ca prove the followg: a) Let C = {( x () ) } N be a strctly creasg cha SP ω (X). Suppose that ( z ) SP ω (X) s a upper boud for C. The there exsts a upper boud ( x ) SP ω (X) for whch ( x ) < ( z ). b) If ( x () ) SP ω (X) for m N the there exsts ( x ) SP ω (X) whch s equvalet to the orm gve by (a ) = max a x (). m A aalogous result for asymptotc structure of spaces wth a shrkg bass s obtaed MT [9] (Proposto 5.). pro3.3 Proposto 3.6. Suppose that (x ) s a ormalzed weakly ull sequece a Baach space X whch has a spreadg model ( x ) whch s ot equvalet to the ut vector bass of l. Assume that belogs to the Krve set of ( x ). The for all sequeces (λ ) R, wth λ ր ad lm /λ = there s a ormalzed block sequece (y ) of (x ) havg a spreadg model (ỹ ) whch satsfes: lm sup / ỹ = lm sup /λ ỹ =. Moreover, the set of all spreadg models, X, whch are ot equvalet to the ut vector bass of l, ad are geerated by weakly ull sequeces, has o maxmal elemet (wth respect to domato). Note that the space X costructed Secto 2 s reflexve ad satsfes the hypothess of the proposto (as does every subspace of X). Proof. Usg lm λ =, choose a subsequece ( k ) of N such that k /λ k 2 k+ k for all k. Sce belogs to the Krve set of ( x ), for every N there exsts a block sequece ) of (x ) whch s detcally dstrbuted wth respect to (x ) ad t has a ormalzed spreadg model ( x () ) as gve Remark.2 rem. (for p = ad ε = ) satsfyg (x () k k 2 x (k). Sce (x ) s weakly ull ad ( x ) ot equvalet to the ut vector bass of l we have that for all N, (x () ) s weakly ull ad ( x () ) s ot equvalet to the ut vector bass of l. We ca also assume wthout loss of geeralty that (x () ) s ormalzed. Let (y ) be the sequece whch s provded by Proposto 3.2 malemma for C k = 2 k. By part (b) of Proposto 3.2 malemma we have that (ỹ ) s ot equvalet to the ut vector bass of l thus lm sup / ỹ = 4

15 by Proposto 2.. prop2.0 Also, by Proposto 3.2 malemma we have that k k 2 k ỹ x (k) k 2. Thus for all k N, = k ỹ /λ k = = k 2 k+ λ k k, whch shows that lm sup ỹ /λ =, = ad fshes the proof of the frst part of Proposto 3.6 pro3.3 oce we ormalze (y ). To prove the moreover part, gve a spreadg model ( z ) SP ω (X) ot equvalet to the ut vector bass of l use the frst part of the Proposto to get (ỹ ) wth (choose λ k = z, for k N) Therefore ( z ) s ot maxmal. lm sup ỹ z =. I some crcumstaces we wll be able to coclude that SP ω (X) admts a trasfte strctly creasg cha. The logcal part of the argumet s a smple proposto. geeral Proposto 3.7. Let X be a separable fte dmesoal Baach space. Let C SP ω (X) be a o-empty set satsfyg the followg two codtos: () C does ot have a maxmal elemet wth respect to domato; () for every ( X ) N C there exsts X C such that X X for every N. The for all α < ω there exsts X (α) C such that f α < β < ω the X (α) < X (β). Proof. We use trasfte ducto. Suppose that X (α) have bee costructed for α < β < ω. The X (β) s chose usg () ad () f β s a successor ordal ad () f β s a lmt ordal. ew Remark 3.8. () The set C = SP ω (X) satsfes codto () by vrtue of Proposto 3.2. malemma Hece f SP ω (X) does ot have a maxmal elemet, the t cotas a ucoutable creasg cha. (2) Suppose SP ω (X) cotas ( x ) such that s the Krve set of ( x ) but ( x ) s ot equvalet to the ut vector bass of l. Let C be the set of all elemets of SP ω (X) whch are ot equvalet to the ut vector bass l. The t satsfes () by Proposto 3.2 malemma ad () by Proposto 3.6. pro3.3 Therefore C cotas a ucoutable creasg cha. Examples of such a space X are the space costructed Secto 2, Gowers-Maurey space GM ([0]) GM ad Schlumprecht s space S ([27]). S2 The followg result s a stregtheg of Proposto 3.6. pro3.3 Frst recall that f (x ) ad (y ) ( N) are two basc sequeces the the bass-dstace betwee them s defed by { d b ((x ), (y ) } ) = sup a x b x : a y = b y =. 5

16 pro3.3a Proposto 3.9. Let (z ) be a ormalzed bass ad C <. Let X be a fte dmesoal Baach space. Assume that for all N there exsts a ormalzed weakly ull sequece (x ) X wth spreadg model ( x () ) such that ( x () ) C-domates (z ) for all N. Assume also that (z ) C-domates ( x () ) for each N. The for every λ ր there exsts a ormalzed weakly ull sequece (y ) X wth spreadg model (ỹ ) so that lm f d b ((ỹ ), (z ) ) λ = 0. Proof. Sce λ ր, we ca choose a sequece ( k ) of tegers such that k2 k λ k for all k. Apply Proposto 3.2 malemma to obta a semormalzed weakly ull sequece (y ) X wth a spreadg model (ỹ ) such that (ỹ ) 2 k -domates ( x k ), for all k N. By part (c) of Proposto 3.2 malemma we also have that there exsts C < such that (z ) C -domates (ỹ ). Let k N ad (b ) k be a sequece of scalars. The k k k b ỹ 2 k b x k 2 k C b z. Thus for k N, f (a ) k ad (b ) k are fte sequeces of scalars satsfyg k a z = k b z =, the k a ỹ k b ỹ C 2 k C = CC 2 k λ k k CC. Hece d b ((ỹ ) k, (z ) k )/λ k k CC whch teds to zero. The result follows by ormalzg (y ). Propostos 3.6 pro3.3 ad 3.9 pro3.3a motvate the followg qst3. Questo 3.0. Whch ormalzed subsymmetrc bases (y ) (f ay) have the followg property: If X s a separable fte dmesoal Baach space so that o spreadg model of X s equvalet to (y ) the there exsts λ ր ad a subspace Y of X such that for all spreadg models ( x ) of ormalzed basc sequeces Y, lm f d b ( ( x ), (y ) ) /λ > 0. Ths questo s a geeralzato of the followg problem rased by Rosethal (whch s solved by Proposto 3.6). pro3.3 qst3. Questo 3.. Let Z be a separable fte dmesoal Baach space so that wheever ( x ) s the spreadg model of a ormalzed basc sequece Z the lm x / = 0. (.e. by Proposto 2., prop2.0 o spreadg model Z s equvalet to the ut vector bass of l ). Does there exst λ ր such that lm λ / = 0 ad for all spreadg models ( x ) of ormalzed basc sequeces Z lm x /λ = 0? 6

17 The questo asks whether all spreadg models of Z must be uformly dstacg themselves from l for large eough dmesos. Questo 3. qst3. ust asks f oe could take (y ) Questo 3.0 qst3. to be the ut vector bass of l. Proposto 3.6 pro3.3 shows that ths s ot true, eve heredtarly. The verso of Questo 3.0 qst3. for the ut vector bass of c 0 s the followg questo. We wll gve a aswer the ext secto. qst3.2 Questo 3.2. Let Z be a separable fte dmesoal Baach space so that wheever ( x ) s a spreadg model of a ormalzed basc sequece Z the lm x =. Ex Does there exst a sequece (λ ) wth λ ր such that for all spreadg models ( x ) of ormalzed basc sequeces Z lm x /λ =? The hypothess of ths questo s equvalet to: o spreadg model of Z s somorphc to c 0. Ideed, suppose ( x ) s a spreadg model of a ormalzed basc sequece (x ) wth lm f x <. We the obta, sce ( x ) s basc, that sup x K for some K <. I partcular, (x ) must be weakly ull ad hece ( x ) s ucodtoal. Thus ( x ) s equvalet to the ut vector bass of c 0. Coversely, f some spreadg model ( x ) s a bass for c 0, the (( x 2+ x 2 )/ x 2+ x 2 ) s equvalet to the ut vector bass of c 0 ad s a spreadg model of ((x 2+ x 2 )/ x 2+ x 2 ). 4. A space havg spreadg models close to c 0 I ths secto we gve a example whch solves questo 3.2 qst3.2 egatvely. T:Ex. Theorem 4.. There s a Baach space X wth a ormalzed bass (e ) so that: a) For every sequece (λ ) (0, ), wth lm λ = there s a subsequece (e k ) of (e ) whch has a spreadg model ( x k ) for whch lm m x /λ m = 0 m b) For every spreadg model ( x ) of a ormalzed block bass (x ) of (e ) lm x =. Before defg X we eed some otato. Let D = =0 {0, } be the dyadc tree ordered by exteso: s = (s ) m t = (t ) ff m ad s = t for m. If s = (s ) m D we set s = m, = 0 ad f s t, [s, t] deotes the segmet {α D : s α t}. A brach β D s a maxmal learly ordered subset. If (β ) {0, }N we wrte β = (β ) to deote the brach (β ) = where β = (β ). tsegmets Lemma 4.2. Let (t ) = be dstct elemets of D. The there exsts a subsequece (t ) of (t ) ad a sequece (s ) D so that s s 2 ad ([s, t ]) = are dsot segmets. 7

18 eq:2 (2) eq:22 (22) Proof. By passg to a subsequece (e.g., usg Ramsey s theorem) we may assume that ether t t 2, whch case we take s = t for all, or t ad t are comparable for all. I the latter case we let s =, t = t ad choose s 2 wth s 2 = t so that {t : s 2 t } s fte. We let t 2 be oe of these t s ad select s 3 wth s 3 = t 2 so that {t : s 3 t } s fte ad proceed ths fasho. Proof of Theorem 4.. T:Ex. For each s D we shall defe a decreasg sequece V s = (V s ()) (0, ]. If s =, V s () = for all. If s = (ε ) m let { : ε =, m} = ( ) k wrtte creasg order. If ε = 0 for m we let V s () = for all. Otherwse for, V s () = /. If < +, set V s () = V s ( ) /( + ). If > k set V s () = V s ( k ). If β = (β ) s a brach, aturally detfed as a sequece of 0 s ad s, V β s defed smlarly. Clearly V β() = for all braches β. If x c 00 (D) we set x = sup V s () x(t ) where the sup s take over all N, ad dsot segmets [s, t ],...,[s, t ] such that s s 2 s. x x follows by cosderg [, t]. The motvato for defg the orm ths maer comes from Lemma 4.2 dsotsegmets ad Case below. The ut vector bass (e α ) α D forms a ormalzed -ucodtoal bass for X, the completo of (c 00 (D), ). We verfy a). Let λ ad choose tegers < 2 < so that λ > 2 < + for all (wth 0 = 0). Let β = for all, β = 0 f / {, 2,...} ad β = (β ). Let β = (β ) = ad let x = e β for N. Let m N. We wll prove that f m ( 0, 0 ] ad 0 < k < < k m the m eq:23 (23) x k 0 +. Thus f ( x ) s ay spreadg model of a subsequece of (x ), by ( eq:2 2), m x λ m < 0 + ( 0 ) 2 ad ths yelds a). Let [s, t ],...,[s, t ] be dsot segmets wth s s 2 s such that for x = m x k, x = V s () x(t ). Sce each V s s a decreasg sequece we may assume that x(t ) 0 for all ad hece m. Also each t s the support of some x kl ad so the segmets must le all o β. I 8

19 partcular whle s < k s possble, s k for 2. Note that m 0 hece by ( 22) eq:22 for 2 the frst elemets of V s are the frst elemets of the sequece ( ) χ [, ], χ (, 2 2 ],, χ (0, 0 0 ]. 0 Thus m x k + 0 = = 0 +, ad ( 23) eq:23 s proved. To see b), let ( x ) be the spreadg model of a ormalzed block bass (x ) of (e α ). By passg to a subsequece of (x ) we have two cases. case Case. There exsts ε > 0 so that x ε for all. I ths case let x (t ) ε for some sequece (t ) D. Passg to a subsequece, usg Lemma 4.2, dsotsegmets we may assume that there exst s s 2 wth ([s, t ]) beg dsot segmets. It follows that for k < < k m m m m x k V sk () x k (t ) ε V sk (). Let β be the brach determed by (s ). Now V β() = by our costructo ad there exsts k 0 so that f k 0 k the V sk () = V β () for m. It follows that m m x ε V β () ad b) holds. case2 Case 2. x 0. Frst ote that there s a fucto δ(m), wth δ(m) 0 as m such that the followg holds: for a arbtrary x c 00 (D) wth x =, cosder dsot segmets [s, t ], [s 2, t 2 ],...,[s k, t k ] wth s s 2 s k, such that k x = V s () x(t ). = The, wheever x δ(m) for some m, the there exsts k k such that s k > m, k > m ad k V s () x(t ) /2. =k Usg ths fact, sce x = for all, ad x 0, we ca costruct ductvely a subsequece (x ) of (x ) (wth = ), ad for all, dsot segmets [s, t ], [s 2, t 2],...,[s k, t k ] wth s s 2 s k, ad tegers k such that s s k t k < s 2 s k t k < s s 2 s k t k ad eq:24 (24) k = x V s (k + ) x (t ) /2 = 9

20 ad such that the sequece k, k +,..., k +k, k 2, k 2 +,..., k 2 +k 2,... s creasg. Let < < m be a creasg sequece. Applyg ( 24) eq:24 for each l, l m ad usg the fact that the sequeces V s (), N are creasg, we get k m l l= = V s l (k + + k l + ) x l (t l ) m/2. Sce all segmets [s l, t l ] are dsot, we deduce m < m 2 l= x l. Hece the spreadg model ( x ) must be equvalet to the ut vector bass of l. Ths completes the proof of b). For all N t s easy to costruct a space X for whch the cardalty SP(X) = SP ω (X) =. Ideed, X = ( l p )2 suffces, where the p s are dstct elemets of (( (, ). Also f 2 < p < p 2 <... the t s ot hard to show that SP ω l )2) p = ω. I ths case oe obtas a fte decreasg cha of spreadg models. But we do ot kow what happes heredtarly. Let us meto some questos (amog may) cocerg the heredtary structure of spreadg models. qst3.3 Questo 4.3. Does there exst a Baach space such that every fte dmesoal subspace there exst ormalzed basc sequeces havg spreadg models equvalet to the ut vector bases of l ad l 2? If such a space exsts, must t cota more (perhaps ucoutably may) mutually o-equvalet spreadg models? More geerally, does there exst X so that for all subspaces Y of X ad p <, the ut vector bass of l p (ad of c 0 ) s equvalet to a spreadg model of Y? Is the space costructed [2] OS or [23] OS2 such a space? I order to aswer Questo 4.3, qst3.3 the aswer to the followg questo may be useful: qst3.4 Questo 4.4. Ca we always somorphcally (or sometrcally) stablze the set of spreadg models by passg to approprate subspaces?.e. for every Baach space X does there exsts a subspace Y such that for every ormalzed basc sequece (y ) Y havg spreadg model (ỹ ) ad for every further subspace Z of Y, there exsts a ormalzed basc sequece (z ) Z havg spreadg model ( z ) such that ( z ) s equvalet (respectvely, sometrc) to (ỹ )? Is the space X costructed secto 2 sec2 a couterexample? qst3.5 Questo 4.5. Let N. Does there exst a Baach space so that every subspace has exactly (somorphcally or sometrcally) dfferet spreadg models? Does there exst a Baach space so that every subspace has coutably ftely may (somorphcally or sometrcally) dfferet spreadg models? May problems are ope cocerg the structure of the partally ordered set SP ω (X) ( the sese of Defto 3.). D3. We state a few of these. qst3.6 Questo 4.6. What are the realzable somorphc structures of the partally ordered set (SP ω (X), )? I partcular, for every fte partally ordered set (P, ) such that ay two elemets admt a least upper boud, does there exst X such that SP ω (X) s somorphc to (P, )? We ote that by Proposto 3.2 malemma ad Remark 3.5, creas f SP ω (X) s fte the oe ca costruct sequeces (ỹ ) ad ( w ) SP ω(x) so that (ỹ ) < (ỹ 2 ) < < ( w 2 ) < ( w ). 20

21 qst3.7 Questo 4.7. Suppose SP ω (X) s fte (or eve coutable). What ca be sad about X? Must some spreadg model be equvalet to the ut vector bass c 0 or l p ( p < )? We address the case SP ω (X) = Secto 5. sec4 sec4 5. Spaces wth a uque spreadg model The followg questo was posed to us by Argyros. qst4. Questo 5.. Let X be a fte dmesoal Baach space so that SP(X) =. Must the uque spreadg model of X be equvalet to the ut vector bass of l p for some p <, or c 0? Oe could also rase smlar questos by restrctg to ether those spreadg models geerated by ormalzed weakly ull basc sequeces or, the case that X has a bass, to those geerated by ormalzed block bases. We gve some partal aswers to these questos usg our techques above. pro4.2 Proposto 5.2. Let X be a fte dmesoal Baach space so that all spreadg models of ormalzed basc sequeces X are equvalet. a) If all the spreadg models are uformly equvalet,.e. f there exsts D R so that the spreadg models of all ormalzed basc sequeces X are D-equvalet, the all spreadg models of X are equvalet to the ut vector bass of l p for some p < or c 0. b) Let (z ) be a ormalzed basc sequece whch domates a (hece every) spreadg model of X. The there exsts C < so that (z ) C-domates ay spreadg model of a ormalzed basc sequece (x ) X. c) If p belogs to the Krve set of the spreadg model ( x ) of some ormalzed basc sequece (x ) of X the ( x ) domates the ut vector bass of l p. d) If belogs to the Krve set of some spreadg model X the all spreadg models are equvalet to the ut vector bass of l. Proof. If X s ot reflexve the there exsts a ormalzed basc sequece (x ) X whch domates the summg bass [2]. J By [25] R (x ) has a subsequece (x k ) whch s ether equvalet to the ut vector bass of l or t s weak-cauchy. I the later case (x 2k+ x 2k ) k s weakly ull ad thus by passg to a subsequece we ca assume that t has a ucodtoal spreadg model whch domates the summg bass ad hece must be equvalet to the ut vector bass of l. Therefore ether case there exsts a spreadg model X equvalet to the ut vector bass of l, ad t s easy to see that a) d) hold. Thus for the proof of a) d) we may assume that X s reflexve. a) Let ( x ) be a spreadg model of X ad let p the Krve set of ( x ). By Remark.2 rem. for every N there exsts a spreadg model ( x () ) of X such that ( x () ) s 2-equvalet to the ut vector bass of l p. Also ( x ) s D-equvalet to ( x() ) thus 2D-equvalet to the ut vector bass of l p. b) Let (z ) be a ormalzed basc sequece whch domates all spreadg models of X. Assume that the statemet s false. The for every N there exsts a ormalzed weakly ) X, havg spreadg model ( x () ), ad there exst scalars (a () ), such that a() x () = 2 2 ad a() z =. By Proposto 3.2 malemma there exsts a semormalzed weakly ull sequece (y ) X, havg spreadg model (ỹ ) such that (ỹ ) 2 ull basc sequece (x ()

22 2 -domates ( x () ) for all. Thus a() ot domate (ỹ ), whch s a cotradcto. ỹ 2 a() x () = 2. Hece (z ) does c) Ths follows from b) ad Remark.2. rem. d) Ths follows from c). rmk4.3 Remark 5.3. If X has a bass (e ) ad the hypothess of Proposto 5.2 pro4.2 s chaged to all spreadg models of ormalzed block bases are equvalet the oe obtas a smlar theorem, whle the coclusos are restrcted to spreadg models geerated by ormalzed block bases. The X s ot reflexve part of the proof s replaced by (e ) s ot shrkg. If the hypothess s chaged to all spreadg models geerated by ormalzed weakly ull basc sequeces are equvalet the oe has two cases: Ether X s a Schur space, hece X R s heredtarly l [25], or X does admt such a spreadg model. Ad the proposto holds the latter case wth the obvous modfcatos. If X s a Baach space for whch all elemets of SP(X) are sometrcally somorphc to each other t follows from Proposto 5.2 pro4.2 that they must all be sometrcally somorphc to l p, for some p <, or to c 0. I the case that p = or the c 0 case, t was show OS [22] that X must cota a copy of l or c 0 respectvely. But the followg questo s stll ope. qst4.2 Questo 5.4. Let < p < ad assume that all elemets of SP(X) are sometrcally somorphc to the ut vector bass of l p. Does X cota a copy of l p? A problem closely related to 4. has bee cosdered by V. Ferecz, A. M. Pelczar ad C. Rosedal [6]: FPR Suppose that X has a bass (e ) for whch every ormalzed block bass has a subsequece equvalet to (e ). Must (e ) be equvalet to the ut vector bass of c 0 or some l p? The authors obta results aalogous to those Proposto 5.2. pro4.2 May addtoal questos rema about the structure of the spreadg models of a Baach space X. sec5 6. Exstece of o-trval operators o subspaces of certa Baach spaces I ths secto we gve suffcet codtos o a Baach space X for the exstece of a subspace Y of X ad a operator T : Y X whch s ot a compact perturbato of a multple of the cluso map. Ths property s related to the log stadg ope problem of whether there exsts a Baach space (of fte dmeso) o whch every operator s a compact perturbato of a multple of the detty. Notce that f a Baach space X cotas a ucodtoal basc sequece the there exsts a subspace Y of X ad a operator T : Y Y such that P(T) s ot a compact perturbato of a multple of the detty for all o-costat polyomals P. Ideed Y ca be take to be the closed lear spa of the ucodtoal basc sequece, ad T a dagoal operator wth ftely may dfferet egevalues, each of of fte multplcty. Gowers [9] G proved that there exsts a subspace Y of the Gowers-Maurey space GM (as defed [0]), GM ad a operator T : Y GM whch s ot a compact perturbato of a multple of the cluso. I [] AS t s show that there exsts a operator o GM whch s ot a compact perturbato of a multple of the detty. It s also kow that some of the asymptotc l ad heredtary decomposable spaces costructed by Argyros ad I. Delya [2] AD admt subspaces o whch o trval 22

23 operator ca be costructed (upublshed work of Argyros ad R. Wager, see also [7] Ga ad Ga2 [8]). Our approach geeralzes the dea of [9]. G Ma3 Theorem 6.. Let X be a Baach space. Assume that there exsts a ormalzed weakly ull basc sequece (x ) X havg spreadg model ( x ) whch s ot equvalet to the ut vector bass of l yet belogs to the Krve set of ( x ). The there exsts a subspace W of X ad a cotuous lear operator T : W W such that p(t) s ot a compact perturbato of a multple of the detty operator o W, for every o-costat polyomal p. The proof uses a coveet auxlary otato. Let F [N] < be a famly of fte subsets of postve tegers. For (a ) c 00 we set { } (a ) l (F) = sup a : F F. Proof of Theorem Ma3 6.. The ma part of the proof s the followg Clam : For every l N {0} there exsts (w (l) ) a semormalzed sequece X, a creasg sequece (M (l+) ) of postve tegers ad a sequece (δ (l+) ) of postve umbers wth δ(l+) <, such that w (0), w (), w (0) 2, w (2), w () 2, w (0) 3,... s a basc sequece X, ad for every (a (l) ) l N {0}, N c 00 ((N {0}) N) we have 34 (25) max l< (a(l) ) l a (l) w (l) (a (l) ) l+, 34a (26) l=0 where for l N ad (a ) c 00 we defe (a ) l = sup N ad where for l, N we set G (l) δ (l) = F (a ) = sup l (G (l) ) N l=0 sup E G (l) = {F N : F M (l) }. Oce Clam s establshed, let W = spa{w (l) map T : W W by Sce (w () δ (l) a, E : l N {0}, N} ad defe a lear T(w (0) ) = 0 ad T(w (l+) ) = 2 l+w(l) for all l N {0} ad N. ) N {0}, N s a basc sequece X, T s well defed. Let (a (l) ) l N {0}, N = a(l) w (l) W. We have c 00 ((N {0}) N) ad x = l=0 Tx = l= = a (l) 2 lw(l ) l= max l< (al ) l x (by ( 25)) l (a(l) ) l (by ( 34 25)) Thus f W deotes the closure of W the T exteds to a bouded operator o W. 23

24 Let p(t) = a t + a t + + a t + a 0 be a o-costat polyomal. We show that p(t) s ot a compact perturbato of a multple of the detty operator I o W. Ideed, for ay {, 2,..., } ad N we have T w () = T 2 w( ) = T w( 2) = = Thus for every scalar λ ad N we have: ( ) (p(t) λi)w () = a T λi w () = Sce w (0), w(), w(0) 2, w(2), w() 2, w(0) 3 < 2 < N such that ( a ( k= + block sequece of w (0), w(), w(0) 2, w(2), w() compact operator. ( a k= + k= + 2 kw( ). ) w ( ) 2 k + (a 0 λ)w ().,... s a semormalzed basc sequece X, there exst )w ( ) 2 k s +(a 0 λ)w () s ) s s a semormalzed,... whch proves that p(t) λi s ot a 2, w(0) 3 Clam follows from Clam 2: There exsts a subspace Y of X wth a bass ad for every l N {0} there exsts a semormalzed weakly ull basc sequece (u (l) ) Y, a creasg sequece (M (l+) ) of postve tegers, ad a sequece (δ (l+) ) of postve umbers wth δ(l+) <, such that the vectors u () for N {0} ad N are dsotly supported wth respect to the bass Y, ad for every l N {0} ad (a (m) ) m {0,,...,l}, N c 00 ({0,,..., l} N) we have that 37 (27) 2 max 39 (28) m l (a(m) ) m l m=0 = a (m) u (m) (/2) l m=0 (a (m) ) m+. (recall that m was defed ( 26)). 34a Oce Clam 2 s establshed, passg for every N to a subsequece of (u () ) (whch does ot affect the estmates ( 27)) 37 ad makg small perturbatos f ecessary, we get (w () ) such that w (0), w (), w (0) 2, w (2), w () 2, w (0) 3,... forms a block bass Y ad for all (a (l) ) l N {0}, N c 00 ((N {0}) N) we have 2 l=0 = a (l) u (l) l=0 = a (l) w (l) 2 Obvously ( 37 27) ad ( 39 28) mply ( 34 25) ad thus Clam follows. l=0 = a (l) u (l). Now we prove Clam 2. We costruct the space Y ad ductvely o l N {0} we costruct the sequeces (u (l) ), (M (l+) ) ad (δ (l+) ) whch satsfy ( 27). 37 The upper ad lower estmates are based o the followg two lemmas of depedet terest, whose proofs we postpoe utl the ed of the secto. oell Lemma 6.2. Let X be a Baach space ad (x ) be a ormalzed weakly ull basc sequece X whch has a spreadg model ( x ) ot equvalet to the ut vector bass of l. The 24

25 for every (δ ) 2 (0, ) there exsts a subsequece (x m ) of (x ) ad a creasg sequece M < M 2 < of tegers, such that for all (a ) c 00 we have (put δ = 2) upperboud (29) a x m sup sup δ a N F N, F M Krv Lemma 6.3. Let X be a Baach space ad (z ) be a ormalzed weakly ull basc sequece X whch has spreadg model ( z ) such that belogs to the Krve set of ( z ). There exsts a subsequece (z ) of (z ) wth the followg property. Gve ay fte subset J N, ay subsequece (M ) of N, ad (δ ) (0, ) wth = δ <, there exsts a semormalzed weakly ull basc sequece (y ) the spa of (z ) J whch s dsotly supported wth respect to (z ) J, such that for all (a ) c 00 ad all y the spa of (z ) J we have K (30) sup δ (a ) l (G ) y + a y. N 37a (3) 37b (32) wth G := {A N : G M } for N. Furthermore, f ( z ) s ot equvalet to the ut vector bass of l the o spreadg model of (y ) s equvalet to the ut vector bass of l. We ow retur to the proof of Clam 2. Sce belogs to the Krve set of ( x ), we ca use Lemma 6.3 Krv ad assume w.l.o.g. that (x ) satfes the cocluso of Lemma 6.3, Krv as stated for (z ). Let K 0, K, K 2,... be dsot fte sets of postve tegers. For all l N {0} we wll costruct dsotly supported u (l) spa{x : K l }, (δ (l) ) N ad (M (l) ) N (satsfyg the codtos as stated Clam 2) so that for all (a (m) ) m {0,,...,l}, N c 00 ({0,,..., l} N) ad y spa(x : s>l K s) we have that l m=0 = 2 max a (m) m l (a(m) (whch yelds ( 37 27) f we put y = 0). u (m) (/2) ) m y + l m=0 l m=0 = F (a (m) ) m+ a (m) u m Costructo of (u (0) ), (δ () ) N ad (M () ) N : Let (δ () ) 2 (0, ) such that 2 δ() <. Sce ( x ) s a spreadg model of (x ) K0 whch s ot equvalet to the ut vector bass of l we may apply Lemma 6.2 oell to obta a subsequece (x m ) of (x ) K0, a creasg sequece (M () ) N of postve tegers, ad δ () > 0 such that for all (a ) c 00 we have 40 (33) a x m 2 (a ) := 2 sup δ () (a ), l (G () ) N where G () = {G N : G M () }. Ths yelds ( 3) 37a for l = 0 whle ( 32) 37b s vacuous. The ductve step - Costructo of (u (l) ), (δ (l+) ) ad (M (l+) ): Assume that we have costructed (u m ), (M (m+) ) ad (δ (m+) ) for m = 0,,...,l so that ( 3) 37a ad ( 32) 37b are 25

26 satsfed whe l s replaced by l. Apply Lemma 6.3 Krv for J = K l, (M ) = (M (l) ) ad (δ ) = (2δ (l) ) to obta a dsotly supported semormalzed weakly ull basc sequece (u (l) ) spa{x : K l } satsfyg: for all (a ) c 00, ad y spa{x : K l } that 42 (34) 2 supδ (l) (a ) y + l a (G (l) ) u (l), N where G (l) = {G N : G M (l) } for N. By passg to a subsequece of (u (l) ) ad relabelg we ca assume that (u (l) ) has a spreadg model (ũ (l) ). By the furthermore part of Lemma 6.3 Krv we have that (ũ (l) ) s ot equvalet to the ut vector bass of l. Let ) 2 (0, ) such that 2 δ(l+) <. Apply Lemma 6.2 oell to obta a subsequece (δ (l+) of (u (l) ) (whch we stll call (u (l) ad δ (l+) > 0 such that for all (a ) c 00 we have 43 (35) a u (l) 2 (a ) l+ := 2 sup ) ), a creasg sequece (M (l+) ) N of postve tegers, N δ (l+) (a ) l (G l+ ), where G (l+) = {G N : G M (l+) } for N. We ow show that ( 3) 37a ad ( 32)are 37b satsfed. Let (a (m) ) m {0,,...,l}, N c 00 ({0,,..., l} N) ad y spa(y : s>l K s). From ( 35) 43 ad the ducto hypothess t follows that l m=0 = a (m) u (m) + a (l) u (l) = l 2 m=0 = l m=0 a (m) u (m) + (a (m) ) m+, a (l) u (l) = whch yelds ( 3). 37a By the ductve hypothess for y replaced by y + estmate y + l m=0 2 max = a(m) m l (a(m) u (m) as follows: ) m y + = a (l) u (l) + l m=0 = a (m) u (m) = a(l) u (l), we ca whch, together wth ( 42 34), mples ( 37b 32). If we are terested oly the costructo of a operator o a subspace whch s ot a compact perturbato of a multple of the cluso map, the the spreadg model assumptos of Theorem 6. Ma3 ca be sgfcatly relaxed ad the argumet would be essetally smpler. Ma2 Theorem 6.4. Let X be a Baach space. Assume that there exst ormalzed weakly ull basc sequeces (x ), (z ) X such that (x ) has spreadg model ( x ) whch s ot equvalet to the ut vector bass of l, ad (z ) has spreadg model ( z ) such that belogs to the Krve set of ( z ). The there exsts a subspace Y of X ad a operator T : Y X whch s ot a compact perturbato of a multple of the cluso map. 26