This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
, Up + 1/q = 1 and -1 < a < 1 . (JoPF = Aa« oo : C$ — Cv is compact}. In this paper we are going to study the set $ endowed with topologies which are related to the Hilbert-Schmidt norm for operators on H2. The main instance is the so-called Hilbert-Schmidt topology, which is defined by the "metric" Lp . In §2 we fix some notation and in §3 we state some general results on the spaces ($,p). Our main results about connected components are contained in §4, and the relationship between the spaces ($,p) and (2,p)-order boundedness of composition operators is investigated in §5. Cv is an injective map from $ to B(HP): in fact, the C^ are even linearly independent in B(HP) (cf. [4]). Recall that an operator T in B(H2) is Hilbert-Schmidt if n>0 belong to -Kp(O). Finally, let p — oo. Implication (a) => (b) is trivial. To see the converse, note that the hypotheses (p ~(oo) xj> and iptp G C(oo)(0) (i.e., H^V'll//00 < 1> by (i)) imply the existence of a constant if > 0 such that 5"1/2 + t"1/2, and so the 2 in terms of Carleson properties of the image measure a^ of a defined by the boundary value function of cp. (2) Applying to our situation well-known results on majorizing and cone absolutely summing operators (cf. [7, IV.3.8]), we may say that C^ is (2,p)-order bounded if and only if (JpC^)* : Lp* —> H2 is cone absolutely summing (p > 1). Note also that this is implied by (JvC — CXQ^ is (2,p)-order bounded, too, whence xo
Proof.- Let us consider G(z) = ]T a n z n
in
Aaq
n=0 oo
and F(z) = ]T b n z n a function in JP a . Using (2.7) for X = LP n=0
46
Blasco: Spaces of vector valued analytic functions and applications
I£ anbnl n=0 JPa-
= I X
Conversely
< C.ll G llo o ll F II = C.ll G l l A o q
let <> | belong to (JP a )* • Take Xn=
II F II
where u n ( z )
= z n and define
G(z)= X^nz n n=0 To verify that
G belongs to AQ a , we shall see that
G belongs to
AaCLQ). Now according to (2.6) we must show that the map CO
OO
T: £PnZ n _> £ x n p n e n n=0
n=0
linear operator from J a into LQ . Now
defines
a bounded
duality
we find a function
f
in the unit ball of
using
LP such that
Hq = I £ *nPnft-n) 1= I <|>( X f(-n)p n u n n=0
n=0
n=0
Hence
(3.7)
II X ^nPne n Hq < II
n=0
Notice that if g(z) = ZJ f(-n)p n z n , then g(z) = hr * f (0) n=0
where
h(z)
OO
= X PnZn and z = re ie . Hence we also have g'(z) = hfr * f(0). n=0 Therefore
M p (g',r) < Mi(h',r) II f llp 1
II g ll J P o <
and consequently 1
J(l-s)aM p (g',s)ds < J(l-s)aMi(h',s)ds = II h IIJO
Combining now this last inequality and (3.7) the proof is completed.
Blasco: Spaces of vector valued analytic functions and applications
47
Remark 3.1. The same proof as in Theorem 3.1 can be done in the vector valued case. Note that we only have applied that L<1 is embedded in (LP)*, what is still true in the vector valued case (see [D-U]). Hence we can see that AaP(X*) is always a dual no matter the geometry of X*, what shows the difference with HP(X*).
REFERENCES. [A-C-P] J.M ANDERSON, J CLUNJE and CH. POMMERENKE On Bloch functions and normal functions. J. Reine Angew. Math. 270 (1974), 12-37. [Bu] A. V. BUKHVALOV, The duals to spaces of analytic vector valued functions and the duality of functors generated by these spaces. Zap. Nauch. Semi. LOMI 92 (1979), 30-50. [D-U] J.DIESTEL and J J . UHL. "Vector Measures " Math, surveys, no 15, Amer. Math. Soc. , Providence, R.I. 1977. [D] P. L. DUREN, 1970.
"Theory of HP spaces". Academic Press, New York ,
[D-S] P.L. DUREN and A.L. SHIELDS. Coefficient multipliers on HP and BP spaces. Pacific J. Math. 32 (1970), 69-78. [D-R-S] P.L. DUREN, B.W. ROMBERG and A.L. SHIELDS. Linear functionals on HP spaces 0 < p < 1 J. Reine Angew. Math. 238 (1969), 32-60. [Fl] T.M. FLETT. On the rate of growth of mean values of holomorphic and harmonic functions. Proc. London Math. Soc. 20 (1970),749-768. [F2] T.M. FLETT. Lipschizt spaces of functions on the circle and the disc; J. Math. Analysis and applic.39 (1972) 125-158. [G] J.B. GARNETT. "Bounded analytic functions." Academic Press. New York 1981 [H-L] G.H. HARDY and J.E. LITTLEWOOD. Some properties of fractional integrals, II. Math. Z. 34 (1932) 403-439. [K] N. KALTON. Analytic functions in non locally convex spaces and applications. Studia Math. 83 (1986) 275-303.
48
Blasco: Spaces of vector valued analytic functions and applications
[S] J.H. SHAPIRO. Mackey topologies, reproducing kernels and diagonal maps on Hardy and Bergman spaces. Duke Math. J. 43(1976), 187-202. [S-W] A.L. SHIELDS and D.L. WILLIAMS. Bounded projection, duality and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162 (1971), 287-302. [T] M. TAIBLESON. On the theory of Lipschizt spaces and distributions on euclidean n-space I,II,III. J. Math. Mech. 13(1964), 407-479, 14(1965),821-839, 15 (1966), 973-981. [Zl] A. ZYGMUND, "Trigonometric Series." Cambrigde Univ. Press, London and New York. 1959. [Z2] A. ZYGMUND, Smooth functions. Duke Math. J. 12(1945), 47-76.
Oscar Blasco Departamento de Matematicas Universidad de Zaragoza Zaragoza-50009 (SPAIN)
AMS Classification (1980): 46E40, 42A45 Key Words: Vector-Valued Analytic Function, problems, Lipschizt-Besov classes, Multipliers.
boundary
values
Notes on approximation properties in separable Banach spaces P . G . CASAZZA AND N . J . KALTON* DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI COLUMBIA, M O . 65211, U.S.A.
(*): The research of the first author was supported by NSF-grant DMS 8702329 and the research of the second author was supported by NSF-grant DMS 8901636 1. Introduction, definitions and discussion of results. Although the example given by Enflo in 1973 [5] settled the approximation problem and the basis problem for Banach spaces, a number of closely related problems have continued to arouse interest. If X is a separable Banach space, there are a number of natural properties intermediate between X having the approximation property and having a basis. Let us first make some definitions. Suppose X is a separable Banach space. Then X has the approximation property (AP) if there is a net of finite-rank operators Ta so that Tax —> x for x £ X, uniformly on compact sets. X is said to have the bounded approximation property (BAP) if this net can be replaced by a sequence T n; alternatively X has (BAP) if there is a sequence of finite-rank operators, T n , such that sup ||Tn|| < oo and Tnx —> x for x £ X. A sequence Tn with these properties will be called an approximating sequence. If X has an approximating sequence Tn with limn_>oo 11TW11 — 1 then X has the metric approximation property (MAP). An important principle [15] that we will use frequently is that if Tn is any approximating sequence for X then there is an approximating sequence Sn satisfying SmSn — Sn whenever m > n and such that for some subsequence Tkn of Tn then limn_oo \\Tkn - Sn\\ = 0. (See Lemma 2.4 of [15]). A slight weakening of the basis property is to require that X has a finite-dimensional decomposition (FDD) i.e. that X has an approximating sequence Tn satisfying TmTn = ^min(m,n) for m,n 6 N. Szarek [24] has given an example to show that a space with an (FDD) need not have a basis. Between (FDD) and (BAP) we can isolate two other natural properties. We say X has the TT—property if X has an approximating sequence of projections, and the commuting bounded approximation property (CBAP) if it has a commuting approximating sequence. We may add to both these properties the corresponding metric properties (TTI) and (CMAP) where we also have limn_>oo \\Tn\\ = 1. In general if X has a commuting approximating sequence Tn with liminf ||Tn|| =Awe say X has A-CBAP. Johnson [12] showed that a space with the (Tr^-property has an (FDD). However it is not known whether every TT—space has an (FDD). The (CBAP) property was first isolated by Rosenthal and Johnson [14] in the early seventies and has most recently been studied by the first author [2]. Let us note that X has A—CBAP if and only if it has an approximating sequence Tn such that
50
Casazza & Kalton: Approximation properties in separable Banach spaces
TmTn = T*min(m>n) for m ^ n, and limsup ||Tn|| < A. (This is doubtless well-known; it is proved in Proposition 2.1 below.) This suggests that it is quite close to the (FDD) property. However very recently Read [22] gave an example of a Banach space with (CBAP) but failing (FDD). Conversely Casazza [2] showed that a space with both TT and (CBAP) has an (FDD), so Read's space is not a TT—space. It is in general not known if (BAP) implies (CBAP). However certain hypotheses on X do give this implication: it holds if X is reflexive or is a separable dual space [13]. Coincidentally, the same hypotheses give that (AP) implies (MAP) (Grothendieck [11]). It has also been shown by Johnson [14] that any space with (CBAP) can be renormed to have (CMAP). These results suggest a close relationship between the properties (CBAP) and (MAP). Our main result is that X has (CBAP) if and only if it can be equivalently normed to have (MAP), so that (CBAP) is the isomorphic version of (MAP). Further if X has (MAP) it has (CMAP). The proofs of these results (Theorem 2.4 and Corollary 2.5) are quite simple modifications of techniques from the study of approximate identities in Banach algebras (due to Sinclair [23]; see also [4]). We also give another condition, the reverse metric approximation property, which implies (CBAP). Pelczynski [21] and Johnson, Rosenthal and Zippin [15] showed that any space with (BAP) is isomorphic to a complemented subspace of a space with a basis. Johnson [14] has shown that if X has (BAP) then there is a reflexive space Y so that X (B Y is a n—space. In fact Y can be taken to be the space Cp (l < p < oo) defined in Section 3. He conjectures that in fact X© Cp has an (FDD) and hence a basis. However, we show that X © Cp has a basis if and only if X has (CBAP). This shows that several possible conjectures are equivalent. We then give some results on renormings of spaces with (CBAP) and conclude by studying spaces X which have an approximating sequence Tn with lim||/ — 2T n|| = 1. This condition is closely related to unconditional forms of the approximation property. We say that X has the unconditional approximation property (UnAP) if there is an
approximating sequence Tn such that if An = Tn — Tn-\ (To — 0) then
where the supremum is taken over all N and all r/t = ±1, i — 1,2,..., N. We introduce the metric version of (UnAP) and relate our work to recent results of Cho, Johnson, Godefroy, P. Saab and Li ([3],[8],[9] and [18]). 2. Equivalent formulations of (CBAP). We will write [A, B] = AB - BA and Y[bj=a Tj = TaTa+1
...Tb.
PROPOSITION 2 . 1 . Suppose X is a separable Banach space and Tn is an approximating sequence for X with TmTn = Tn for m > n. Let \ = liminf ||T n ||. If £ ||[Tn, T n +i]|| < oo, then X has X—CBAP (and, further has an approximating sequence Rn for which
RmRn = #min(m,n) form ^ n and limsup ||J2»|| < A.) PROOF: We first show that we can suppose that Tn(X) = T%{X). Let Pn be any bounded projection of X onto Tn(X) and choose a sequence 0 < an < 1, so that
Casazza & Kalton: Approximation properties in separable Banach spaces
51
^ a n | | P n | | < oo and —a n /(l — an) is not an eigenvalue of Tn. Then we may replace Tn by (1 — an)Tn + anPn and the hypotheses of the Proposition will still hold with the additional constraint that Tn(X) = T%(X). Now let en = ||[r n ,T n + 1 ]||. For n G N and k > 1 we define A(n,k) = U"=n~X TjLet .4(n,0) = /. Then for k > 1, A(n,* + 1) = A(n,fc) + A(n,fc -
l)[Tn+k_uTn+k).
Thus if Mn(k) =maxi<j
It now follows that A(n, k) is norm-convergent and we can define Sn = flyln ^V = limjt^oo A(n, A;). Clearly the operators Sn are finite-rank with Sn(X) = S%(X) = T n (X). Further if m > n we have SmSn = £ n and 5 n is an approximating sequence with ||Sn|| < Pn\\Tn\\ where lim n _oo^ n = 1. A simple calculation also shows that if m > n then
so that [Sm9Sn] = Sn{I-Sm)
=
A{n9m-n
Thus for a suitable constant K, independent of m, n we have
whenever m > n. In particular we have limm_,oo ||[^m> Sn]\\ = 0 for each fixed n. We now pass to a subsequence Vn of Sn so that limsup ||Vn|| < A. We will also have VmVn = VnioTm>n and lim^oo ||[Vm, Vn]|| = 0 for each n. Finally, by induction we will choose an increasing sequence of positive integers (n^) and operators Rk so that: (1) RUX)=Rk{X)=Vnk{X). (2) Rk is a polynomial in Vni,..., Vnk. (3) For 1 < / < k, RkRi = RtRk = Rh (4) p*||<||S n J| + 2-*. To start let n\ = 1 and R\ =V\. Now suppose rt\,..., nk and R\,..., Rk have been determined to satisfy (1-4). Since Rk(X) = Rk(X) we can find an operator Wk which is a polynomial in Rk so that WkRkx = x for x G .Rjfc(X). Now, limy^j^oo ||J?fc(/ —Vm)|| = 0 since Rk is a polynomial in V n i , . . . , Vnk. Thus we may pick rijk+1 so that
We then define Rk+i =Vnk+1 +WkRk(I
-Vnk+1).
52
Casazza & Kalton: Approximation properties in separable Banach spaces
Clearly conditions (2) and (4) above hold. For condition (3) note that / — (/ - WkRk)(I - Vnjk+1) from which it follows that Rk(I - Rk+1) = {I - Rk+i)Rk = 0 or RkRk+i = Rk+iRk = Rk. Now if 1 < / < k then RiRk+i = RiRkRk+i = RtRk = Rt and similarly Rk+iRi = R\. Thus (3) is verified. For (1) we clearly have Rk+i(X) C Vnk+1(X). It suffices to show that .Rfc+i is injective on Vnk+1(X). Indeed suppose x G Vnk+l(X) and Rk+\x = 0. Then since WkRk(I - Vnk+1)x G Rk{X) = Vnk(X) we have Vnk+1x e Vnk(X). Thus V*k+lx = Vnk+lx and by the fact that Vnic+1 is injective on Vnk+1(X) we have x = Vnk+1x and hence x = 0 as required. It is now immediate that X has A—CBAP." COROLLARY 2 . 2 . Suppose X has an approximating sequence Tn for which
lim ||[T ro ,r n ]||=0 and liminf^oo ||T n|| = A. Then X has X-CBAP. REMARK: By lim^n—.oo amn = 0 we mean that given e > 0 there exists N so that if m,n > N then |a m n | < e. PROOF: We may find a subsequence Tnk andfiniterank operators Sk so that lim* ||Tn/t || = A, US* - Tnk || -> 0, and S^ = Sk for / > *. Then lim m>n ^ oo ||[S m , S n ]|| = 0 and so by passing to a further subsequence we can apply the Proposition. COROLLARY 2.3. Suppose X has an approximating sequence Tn for which lim ||[T m ,T n ]||=0 m—»-oo
for each fixed n and liminf ||r n || = A. Then X has X-CBAP. This Corollary is immediate from the Proposition. The same result without the precise estimate on the constant was shown in [2]. We now come to our main result. The argument in the next theorem is a simple modification of a theorem of Sinclair [23] on approximate identities in Banach algebras. Sinclair shows that if A is a Banach algebra with a bounded two-sided sequential approximate identity then it has a commuting approximate identity with the same bound. This result can be applied directly to the algebra K (X) of compact operators on X when X* is separable and has (AP), and hence (MAP). Under these circumstances K{X) has a norm-one two-sided approximate identity, and we can recover Theorem 2 of [2], In general, however, some modification of Sinclair's approach is necessary. THEOREM 2.4. Suppose X is a separable Banach space with (MAP). Then X has (CMAP). PROOF: We shall suppose that X has an approximating sequence Tn with T m T n = Tn for m > n and ||T*n|| < 1 + en where J2 en = P < oo. For t > 0 define the operators
Vn(t) = e~nt exp(( £ Tk) = e~nt f^t(T1 k=l
3=0 •''
+
- + Tn)>.
Casazza & Kalton: Approximation properties in separable Banach spaces
53
Then
Let En = Tn(X). Then each En is an invariant subspace for every T m and hence also for every Vm[t). Rewriting Vm(t) as exp(* £ £ ^ ( 2 * ~ J)) & i s c l e a r t h a t i f x € En and m > n then Vm(t)x = Vn(t)x. It follows therefore from the bound on the norms of Vn(t) that we can define S(t) by S(t)x = limn_+oo Vn(t)x for all x G X. Clearly \\S(t)\\ < eP*. Furthermore S(t) has the semigroup property S(*i + t2) = S(ti)S(t2) since each Vn(t) is a semigroup and the property is preserved by strong limits. We further claim that each S(t) is compact for t > 0. Indeed suppose / G N and that x e En where n > I. Then d(S(t)x,Ei) = d(Vn(t)x,Ei). It is then easy to see, by expansion, that the operator exp t(Ti H h Tn) — exp t(Ti+i -\ h Tn) has range contained in E\. Thus d(S(t)x, Ei) = e-ntd(exp t(Tl+1 + • • • + Tn)x, Ex) <e-^||ex <e-n*ex < epte-lt\\x\\. Hence for all x £ X, d{S(t)x,Ei)
54
Casazza & Kalton: Approximation properties in separable Banach spaces
Now (Yl TJj)(Ti -\ h Tn) consists of n terms of which n — 1 have at most the same weight w — w(l\,- - •,/ n ) and one has weight w + 1. Thus if if we define Wk by
where the sum is over all terms Yl TJ' in (J2 Tj)k then Wk+i < (n — 1 + \)Wk for every k. As Wi < n - 1 + A we obtain Wk < (n - 1 + A)fc for k > 1. Thus
and so Thus repeating the proof of Theorem 2.4 we obtain
The proofs now goes through unchanged since ||£(*)|| < Ae*^"1) and limt_.o Hence X has A-CBARWe will use Theorem 2.6 to give another characterization of (CBAP). We say that X has the reverse monotone approximation property (RMAP) if there is an approximating sequence Tn with linin-H-oo ||7—Tn \\ = 1. We first prove a simple equivalence for (RMAP). PROPOSITION 2.7. (i) X has (MAP) if and only if there exists a > 0 and an approximating sequence (Tn) with lim ||7 + aT n || = 1 + a. (ii) X has (RMAP) if and only if there exists a > 0 and an approximating sequence (Tn) with\im\\I-aTn\\ = l. PROOF: The proof of (i) is again somewhat similar to the proof of Theorem 2.3. By passing to a subsequence one may suppose that ||7 -f- aT n || < 1 + a(l -f en) where ^2 en = f3 < oo. Then, defining Vn(t) as in Theorem 2.3 one obtains, by estimating II exp(* £fc=i(J + «^fe))||, that ||Vn(crf)|| < eaf3t and the proof goes through as before. For (ii) it suffices to consider the case a < 1 by a simple convexity argument. We may further suppose TmTn = Tn for m > n. Then pick a sequence of integers ln so that lim/n = oo and lim \\(I - aTn)l» \\ = 1. Then set Sn = I - (I - ctTn)ln. Clearly (Sn) is an approximating sequence with lim ||7 — 5 n || = l.» THEOREM 2.8. Let X be a separable Banach space with (RMAP). Then X has (CBAP). To prove Theorem 2.8 we require the following lemma: LEMMA 2.9. Suppose X has (RMAP). Then X has an approximating sequence Tn with lim || J - T n || = 1 and limsup ||(Tn - Tn2)2|| < If. PROOF: We assume Sn is an approximating sequence with SmSn for m > n and \\I ~ Sn\\ < 1 + £n where en | 0. Put Tn = £ Ej3=n+i Sj- T h e n t h e properties of Tn specified in the lemma are clear except possibly for the last.
Casazza & Kalton: Approximation properties in separable Banach spaces
55
Consider
i=n+l.?=n+l Zn Zn
Zn
Zn
t=n+l j=n+l Jfc=n+1 /=n
Now (/ - Ti)Tj(I - Tk)Ti vanishes if either i > j or k > /; this eliminates all but n2(2n -f I) 2 terms. Consider those remaining terms where k < I < 2n < i < j . In this case (/ - Ti)TjTkTi = (I - Ti)TkTi = 0 and hence the term becomes (/ - T^TjTi = 0. There are \n2(n — I) 2 such terms. Thus there remain at most n 2 (2n+1) 2 — \n2(n— I) 2 terms of norm at most (2 + e n ) 2 (l + c n ) 2 . Hence
||(T n - Tn2)2|| < -L( Thus
PROOF OF THEOREM 2.8: We may suppose X has an approximating sequence Tn with | | / - r n | | < l + €n where 11(1 + ^ ) < 2, TmTn = Tn for m > n and ||(Tn - T 2 ) 2 | j < 19/20 for all n. This is possible by the lemma. We then define a new approximating sequence (Sn) by
I-Sn=(l[(I-TkA(I-Tn). \k=i J Thus || J - 5 n || < 4 and so ||5 n || < 5. Now if An = I — Sn we have, provided ln > 1,
k=l
\k=l
]
Thus if p i , . . . ,p n -i are any polynomials,
(j[Pk(Ak)\ An = fiflW - r*)](/ - Tfc)2^) (/ - Tn). In particular if i\ < i^ < • • • < i m = n,
\fc=i
where (3k = 1 if k G {i'i,... ,« m -i} and /3k = 0 otherwise. Hence
ft*
k=i
56
Casazza & Kalton: Approximation properties in separable Banach spaces
This implies an estimate, since ||/ + 5»m|| < 6, m
m— 1
k=l
k=l
^Q
from which we obtain
A:=0
Notice that S ^ X ) C T n (X); hence if m > n, (J - Sm)Sn = 0 so that 5 m 5 n = 5 n Hence S^S^ = 5^ and so we may apply Theorem 2.6 to to the approximating sequence Si to deduce that X has (CBAP).-
3. Complementation and renormings. PROPOSITION 3 . 1 . Suppose X is a separable Banach space and Y is a separable reflexive Banach space so that X 0 Y has (CBAP). Then X has (CBAP). PROOF: Suppose Sn is an approximating sequence for X ®Y such that SmSn = £min(m,n) f° r m ¥" n- Let P be the projection onto X. Consider the operators PSn : Y —• X. Then for every t / G F w e have lim||PS n y|| = 0. Hence since Y is reflexive we have lim(PS n )*z* = 0 weakly for every x* e X*. Thus ([17]) PSn converges weakly to zero in £(y, X). Hence if Q = I- P, PSnQ converges weakly to zero in £ ( I © Y). Now we may pass to a sequence of convex combinations Rn which is still an approximating sequence for X 0 Y and so that lim ||i\R n Q|| = 0. Define Tn : X -> X by Tnx = PRnx. Then Tn is an approximating sequence for X and [Tm,Tn] = PRnQRmP ~ PRmQRnP \x • Hence limm>n_+00 ||[r m ,T n ]|| = 0 and the result follows by Corollary 2.2.Our next result is a slight modification of an argument in [2]. PROPOSITION 3.2. Let X be a separable Banach space with an approximating sequence Tn such TmTn = r m i n ( m , n ) for m ± n. Let En = (Tn - T%){X). Then for 1 < p < oo, X 0 lp(En) has an FDD. Furthermore if we denote by Sn the associated partial sum projections, we have
k=l
PROOF: We use an argument which dates back to Johnson [13] and is exploited in [2]. Define projections 5 n o n l 0 Lp(En) by
Casazza & Kalton: Approximation properties in separable Banach spaces
57
Then SmSn = Smin(m,n) and Sn is an approximating sequence so that X © tp(En) has an (FDD). For x € X Tnx - Snx = (0,0,... ,0, {Tn - T n ) 2 x,0,0,...) where the only non-zero entry is in the position of En. Since p > 1 the last part follows easily." Let Cp denote the space lp{Fn) where Fn is a sequence of finite-dimensional Banach spaces dense in the Banach-Mazur sense in the collection of allfinite-dimensionalBanach spaces (we may assume each space is repeated infinitely often). This space has been studied extensively by Johnson and Zippin (see [16]); it is noted by Johnson [13] that X @CP has an (FDD) if and only if X © Cp has a basis. The next Corollary combines the two preceding results. COROLLARY 3.3. Let X be a, separable Banach space and suppose 1 < p < oo. Then X has (CBAP) if and only if X © Cp has a basis. We remark that Lusky [20] has shown that if C^ denotes co(Fn) then X has (BAP) if and only if X © C^ has a basis. Let us note a brief proof of Lusky's theorem. Let Tn be an approximating sequence for X with TmTn = Tn for m > n and let En = Tn(X). We define Sn : X © co(En) -+ X © co{En) by
where zk = Tk{I - Tn)x - Tkyn + yk for 1 < k < n and zk = 0 for k > n + 1. Then SmSn = 5min(m>n) and so X © co(En) has an (FDD). Thus X © C^ has an (FDD) and hence also a basis. We now apply the complementation results to give a renorming theorem. We require first the following lemma. LEMMA 3.4. Suppose X has an (FDD) with partial sum operators Sn, and suppose 0 < a < 2. Then X can be equivalently renormed so that \\Sn\\ = ||7 — a5 n || = 1 for all n. PROOF: It suffices to show that the semigroup of operators generated by Sn, I — aS n , n G N is bounded. To do this it suffices to consider a product
T=f[(I-aSik) k=l
where i\ < i2 < • • • < in- We can rewrite I — a 5 m = I — Sm + /?5 m where /? = 1 — a. Then k=i
where S{0 is defined to be zero. Thus
58
Casazza & Kalton: Approximation properties in separable Banach spaces
where M = sup n \\Sn\\.m THEOREM 3.5. Suppose X has (CBAP) and 1 < a < 2. Then X can renormed so it has a commuting approximating sequence (Tn) with
lim ||r«||= lim | | / - a r n | | = l n—>oo
n—KX)
and
limsup||rn -rn2 ||oo
^
In particular X can be renormed to have both (MAP) and (RMAP). PROOF: We use Proposition 3.2. We can find an (FDD) of a space X®Y with partial sum projections Sn and a commuting approximating sequence Tn for X so that
Here we have replaced the original sequence Tn by its sequence of Cesaro means. By Lemma 3.4 we can renorm X © Y so that ||5 n || = ||/ — aS n || = 1 for all n. Then under the same renorming restricted to X we easily get that the first equation holds for the sequence Tn. Further more if Rn = ^ ]Cfc=i ^*>
so that 2i
Thus lim sup | | r n - T , J | | < i . . We now demonstrate the limits of this renorming by using a simple modification of an argument of Beauzamy [1] and Esterle [6], PROPOSITION 3.6. Suppose X is a Banach space and that T is a bounded operator on X. Suppose \\T - T2\\ = 6 < \. Then there is a projection P on X such that {x:Tx = x}c P{X) C T{X) and
Casazza & Kalton: Approximation properties in separable Banach spaces
59
PROOF: Define
and P=\{I-{I-2T)S). Since (1 — 4z)~2 has a power series expansion ]Cm=o Cm)2™ v a lid for |^| < ^, it is clear that | | 5 | | < (1 - 40)~ 2 and by a power series manipulation that (/ - 2T)2S2 = I. Hence P is a projection on X and the estimate on ||P|| follows. Note also that
P = ZT2 - 2T3 + - ( / - 2T) £ The remaining properties of P follow easily." A result of Casazza [2] asserts that a separable Banach space has an (FDD) if and only if it has (CBAP) and the 7r-property. In view of Proposition 3.6 we obtain: THEOREM 3 . 7 . Let X be a separable Banach space. Suppose X has an approximating sequence Tn for which l i m s u p , ^ ^ \\Tn — T2\\ < \. Then X has the n-property, and if X has (CBAP) then X has an (FDD). We remark that Read [22] gives an example of a reflexive Banach space with (CBAP) but having no (FDD). Thus this corollary shows that Read's space cannot be renormed to have an approximating sequence Tn for which limsup ||T n — T2\\ < \. Motivated by Theorem 3.5 we introduce the unconditional metric approximation property (UMAP). We shall say that X has (UMAP) provided it has an approximating sequence Tn for which limn_>oo ||/ —2Tn|| = 1. The justification for this terminology lies in the following: THEOREM 3 . 8 . A separable Banach space X has (UMAP) if and only if for every e > 0 there exists an approximating sequence (Tn) so that if An = Tn — T n _i for n G N (with To = 0) then for every N G N and rn = ± 1 , i = 1,2,..., N then N
t=l
PROOF: First suppose X has (UMAP), and e > 0. Then X has an approximating sequence Tn for which TmTn = Tn for m > n and | | / - 2 r n | | = l + 6n where I K 1 +^n) < 1 + e. Defining An = Tn - Tn-i as above with To = 0 we have for N e N and nt: = ± 1 , N-l
VN J Q (/ — (1 — rjN^iT)N-i-i)TN-i) t=1
N-l
— "HN JQ (/ — Tjv-i + t=1 N-l
r)N-i'nN_i_1TN_i)
60
Casazza & Kalton: Approximation properties in separable Banach spaces
Thus N
N-l
|| £ ViAi\\ = \\TN J ] (/ - (1 - 77N-t -^-i-i)^_ t)|| 1=1
1=1
For the converse direction suppose (Tn) is an approximating sequence for which for every N and rji = ±1 we have N
i X>A,-II < i + «• t=l
Then for any n and m > n
t=n+l
*=1
so that ||J - 2Tn|| < liminfm_oo ||Tm - 2Tn|| < 1 + e. It follows easily that X has an approximating sequence Sn for which ||7 — 25 n || —• l.« Let us say that X has UCMAP if for every e > 0 there is a commuting approximating sequence Tn for which if An = Tn — Tn_i and rji = ±1, i — 1,2,..., N then IIEi=i»?M.-||
Casazza & Kalton: Approximation properties in separable Banach spaces
61
so that \\Rn — Vn\\ —•> 0. Then Rn is a commuting approximating sequence for which ||/ - 2Rn\\ -> 1 and the argument of Theorem 3.8 shows that X has (UCMAP). Now let us show (i) => (iv). Let Tn be an approximating sequence for X such that lim||/ - 2Tn|| = 1. For € C{X)* we define 11(0) G £(X)* by U{
4. Concluding remarks. We first make some comments on the example of Read [22]. Read shows that there is a subspace X of C2 so that X has (BAP) but no (FDD). This answers a question raised by Johnson and Zippin [16], (cf. also [19], p. 86) as to whether every subspace of Ci with the approximation property is of the form l2{En) with En finite-dimensional. Clearly X is reflexive and has (CBAP) and even (UCMAP) by Theorem 3.9. It follows by Corollary 3.3 that X ® C^ has a basis. Hence by the results of Johnson-Zippin [16] ([19] p.85), X@C2 is of the form ^(En) and hence X© C2 is isomorphic to C2. Hence X is isomorphic to a complemented subspace of C2. Thus we have an example of a complemented subspace of a space with a (UFDD) which fails to have an (FDD). The major remaining open problem here seems to be whether (BAP) implies (CBAP). Let us note that this is equivalent to the problem of whether a n—space has an (FDD). Indeed if X has (BAP) but not (CBAP) then for any 1 < p < oo X© Cp is a 7T—space (cf. Johnson [13]) but must fail (CBAP) by Theorem 3.1. For the converse, Casazza [2] shows that a n—space with (CBAP) has an (FDD). Yet another form of this problem [13] is whether for any X with (BAP) X © Cp has a basis (when 1 < p < oo.) A related problem is whether (UnAP) implies (CBAP). We also may raise the question of whether (UnAP) and (CBAP) together will imply the existence of a commuting approximating family (Tn) for which the series S ( T n — T n _i) is weakly unconditionally Cauchy. In a similar vein, does (UMAP) imply (UCMAP) in general? This is proved for reflexive spaces in Theorem 3.9; obviously (UMAP) implies (CBAP).
62
Casazza & Kalton: Approximation properties in separable Banach spaces
References. 1. B. Beauzamy, Introduction to operator theory and invariant subspaces, North Holland, 1989. 2. P. G. Casazza, The commuting B.A.P. for Banach spaces, Analysis at Urbana II, (E. Berkson, N.T. Peck and J.J. Uhl, editors) London Math. Soc. Lecture Notes, 138 (1989) 108-127. 3. C.-M. Cho and W.B. Johnson, A characterization of subspaces X of lp for which K{X) is an M-ideal in L(X), Proc. Amer. Math. Soc. 93 (1985) 466-470. 4. R.S. Doran and J. Wichmann, Approximate identities and factorization in Banach modules, Springer Lecture Notes 768, Berlin 1979 5. P. Enflo, A counterexample to the approximation property in Banach spaces, Acta Math. 130 (1973) 309-317. 6. J. Esterle, Quasi-multipliers, representations of H°°y and the closed ideal problem for commutative Banach algebras, Springer Lecture Notes No. 975, (1983) 66-162 7. M. Feder, On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 24 (1980) 196-205. 8. G. Godefroy and D. Li, Banach spaces which are M-ideals in their biduals have property (u), Ann. Inst. Fourier (Grenoble) 39 (1989) 361-371. 9. G. Godefroy and P. Saab, Weakly unconditionally converging series in M-ideals, Math. Scand. 64 (1989) 307-318. 10. G. Godefroy and P.D. Saphar, Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32 (1988) 672-695. 11. A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Memoir Amer. Math. Soc. No. 16, 1955. 12. W.B. Johnson, Finite-dimensional Schauder decompositions in TT\ and dual TZ\ spaces, Illinois J. Math. 14 (1970) 642-647. 13. W.B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971) 337-345. 14. W.B. Johnson, A complementably universal conjugate Banach space and its relation to the approximation property, Israel J. Math. 13 (1972) 301-310. 15. W.B. Johnson, H.P. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971) 488-506. 16. W.B. Johnson and M. Zippin, On subspaces and quotients of {Yl^n)tp and (EGn)co, Israel J. Math. 13 (1972) 311-316. 17. N.J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974) 267-278. 18. D. Li, Quantitative unconditionality of Banach spaces E for which K(E) is an M-ideal in £(E), Studia Math 96 (1990) 39-50. 19. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: sequence spaces Springer, Berlin-Heidelberg-New York 1977. 20. W. Lusky, A note on Banach spaces containing c0 or C ^ , J. Functional Analysis 62 (1985) 1-7. 21. A. Pelczynski, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971) 239-242. 22. C.J. Read, Different forms of the approximation property, to appear
Casazza & Kalton: Approximation properties in separable Banach spaces
63
23. A.M. Sinclair, Bounded approximate identities, factorization and a convolution algebra, J. Functional Analysis, 29 (1978) 308-318. 24. S.J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987) 81-98.
Moduli of Complex Convexity WILLIAM J. DAVIS
Department of Mathematics The Ohio State University Columbus, OH 43210, USA Abstract. In our original paper on complex uniform convexity, the author, D. J. H. Garling and N. Tomczak-Jaegermann introduced several possible moduli of complex convexity. The only moduli studied in that paper were the // p (e)'s. Remarks were made concerning two other moduli, the four point modulus and the tangential modulus. No proofs were given for the assertions about these moduli which appeared in that paper. Here we remedy that situation by giving those proofs.
In [D-G-J], a study of complex notions of uniform convexity was begun. In particular, a (complex) quasi-normed linear space E is said to be p — uniformly — PLconvex (0 < p < oo) if the moduli
Hp(e) = inf{(i- [^ ||* + euy\\'d0)i - 1 : ||«|| = 1, ||y|| = e} ^ Jo for p < oo, (respectively, H^e) = inf {sup ||x + e"y|| - 1 : ||z|| = 1, ||y|| = e} 0
for p = oo) are positive for positive e. It was then shown, for example, that for p in the range 0 < p < oo, these moduli are equivalent, and so the space E is said to be uniformlyPL — convex if it is p — uniformly — PL — convex for any p in (0, oo), and uniformlyHoo — convex in the last case. The study began with the examination of the results of J. Globevnik [Gl] in which, rather than examining moduli defined on complex circles of the form {x -f et9y : 0 € T}, the moduli were defined using only the four points {x -f- iky : k = 0,1,2,3}. We shall show first that the moduli defined in this way are equivalent to the Hp moduli above. In the case of real Banach spaces, and real moduli of convexity, it is natural to consider the moduli defined by line segments tangent to the ball of the space as well as the usual definitions using mid points of segments whose endpoints lie on the sphere. In the complex case, it is impossible in general to ask that all of the points x + etOy lie on the surface of the sphere. However, it is possible to examine moduli of complex convexity defined by pairs £,?/, as above, with the added restriction that the vector y is tangent to the sphere at the point x. The interesting thing about these moduli is that they are all equivalent in the range 0 < p < oo and that they are all equivalent not to Hp with p finite, but with
66
Davis: Moduli of complex convexity
One of the problems with the concepts of complex uniform convexity defined in [D-G-J] has been the difficulty of establishing ties, with that definition, with other concepts such as the analytic Radon-Nikodym property, with complex martingale convergence theorems and with renorming theorems in the spirit of the Enflo-Pisier [E,P] theorems in the real case. Recent work on such questions has led to exciting advances, mainly in the direction of complex martingales such as the Hardy martingales of Garling [G] (see also the work of Xu [X]). There are, for example, in [X], more severe moduli defined which might also be compared with those of [D-G-J] as given above. We shall see easily that they are also comparable in the same way with their four-point and tangential counterparts.In looking at them we may find the essential difference between uniform PL-convexity and Hardy convexity. Throughout this note, we shall assume that the spaces involved are infinite dimensional, or at least of dimension large enough to render our definitions meaningful. Definitions: Here we recall the relevant definitions of the moduli we shall compare. First, for the four-point moduli, we define
for 0 < p < oo, and with the obvious change for p = oo. For the tangential moduli, we define Tp(e) exactly as we defined Hp(e), except that we restrict y's that are used to being tangent to the ball of E at x. Before we define the analogous moduli for the Hardy versions of complex convexity, let us recall some definitions. If we let T denote the unit circle in the complex plane with its Haar measure, a, and if we denote by T N the product of countably many copies of the circle, we can define Analytic Martingales :
fc=0
where <^>n(®) = <^n(#i,02, • • • ,#n-i)- Notice that these are nothing more than martingale transforms of the Steinhaus sequence {el6n}. Due to inherent difficulties of proving convergence of such martingales, and of relating the boundedness of them to the complex geometric structure, Garling [G] defined a broader class of such martingales, the Hardy Martingales as k=0
where now
*»(©)= ]£
The corresponding complex moduli are then defined as = inf{( / ||x + P{6)\\da) -
Davis: Moduli of complex convexity
67
where P{6) = X3n>i xnCin0 is a trigonometric polynomial with values in E. Then, as space E is called uniformly Hp convex if these moduli are positive for positive c. The advantage of this stronger definition is that blocking techniques may be used with the Hardy martingales to establish the links with the geometric theory which were missing in the previous instances. As with the Hp moduli, we define the analogous four-point and tangential moduli, now called Cp(e) and tp(e) in exact correspondence with these new Hardy moduli. At this time, we do not see the equivalence with their capitalized counterparts. Part of the difference lies in the fact that in the equivalence of, for example, Cp with Hpy only convexity is used. In attempting to establish equivalence for cp and hpj it appears that subharmonicity must be used, and similar arguments must fail due to the fact that purely atomic measures cannot be Jensen measures. Proofs of the equivalence. In this section, we shall prove the equivalences announced above for the moduli Tp and Cp. We shall also indicate the changes in proof necessary to handle their lower-case counterparts. THEOREM 1. Let E be a Banach space. For 0 < p < oo, the moduli Tp(e) are equivalent. Further, they are all equivalent to the modulus HQQ. PllOOF: Let x,y € E, \\x\\ = 1, ||y|| = c. Since 2irtf«fo *s a probability measure, it is clear Thus we need only demonstrate that (/ T ||x + eiey\\Pda)p > (/ T \\x + ei$y\\qd
Too(e) > Tp(e) >
AT^)
for all pe (0,co). Now we must check the equivalence of the Tp's with H^. First, we must recall that Hoo(e) = o(e) in infinite dimensional spaces. Select x,y € E with ||x|| = l,||y|| = c such that \\x + y\\ = sup0\\x + eiey\\ < 1 + 2Hoo(e). Let / € J5?M|/|| = 1 = f(x). Then we have that ||x + y|| > 1 + |/(y)|, so that |/(y)| < 2Hoo(e) < rje, where n's smallness will become apparent. In the linear span of x and y in E, let x + u(d) denote the intersections of the lines through x + ei9y with the (complex) line [/ = 1], If n is small enough, we have f < IK*)|| < 2c. We see, then, that for all 0, ||x + u(0)\\ < \\x + ei$y\\(l + 2Hoo(e)) < (1 + 2Hoo{e)f. It follows that T ^ f ) < hH^ie). D Now we move to the equivalences of the moduli Hp and Cp.
68
Davis: Moduli of complex convexity
THEOREM 2. For 1 < p < oo, we have Hp(e) > Cp(e) >
Hp(^).
PROOF: The case of p = oo is rather easy: Let ||x|| = 1, ||y|| = e and consider the circles {x + eiey},{x + -j-ei6y) and the square co{x + iky : k = 0,1,2,3}. Clearly we have |a: + el*y|| > max* \\x -f i*y ||, and simple convexity gives us that max* ||a; -f **y|| > \x + -7«el^y||. Letting x and y range over all suitable choices proves the equivalence of HQO and
C^.
The case for 1 < p < oo is true for the same reasons, but demands some further explanation. It is easiest to see for the case p = 1, and is not essentially different for the other values of p, so we present that case only. Again, let x and y be suitably chosen. Then,
x + eidy\\da > f [I £ \\x + ikei9y\\]da > nun \j2\\x
+ ikei6y\\.
It follows that Hi(e) > C\{e). For the second part, notice that each point on the circle {.T + -j^e%ey} can be written as a convex combination of the four points {x+iky : 0 < k < 3} as follows:
Thus,
' \x + ei9y\\d*<J2£\k(0)\\* + i' s\'mmetry, and the fact that 5^ A*(0) = 1, we get Xk(O)da = i
0< k <3
which proves that H\(-j~) < C\{e). D
Comments on the Hardy moduli: Let us look at the Hardy modulus, h\{e). In its definition, the infimum is taken over trigonometric polynomials of the form p(@) = S/i>i xnGtn6 with UPHLXCT) > e- A key element of the proof of the equivalence of Cp with Hp was the fact that convexity allows us to estimate \\x + ret0y\\LE(T) above b}^ ||.?; + GlOy\\Lx{ii) f° r any of the four point supported probability measures one gets by rotating the natural one. Since no four point measure can be a Jensen measure, it is clear that one cannot extend that equivalence proof as it stands to the case of the Hardy moduli. The question about the equivalence of the various definitions of complex uniform convexity remains open. It may very well be that the key to establishing that they are different lies in the above sort of observation. This line of investigation is worthy of pursuit.
Davis: Moduli of complex convexity
69
REFERENCES
[D-G-J] W. J. Davis, D. J. H. Garling and N. Tomczak-Jaegermann, The complex convexity of quasi— normed spaces, J. Funct. Anal. 55 (1984), 110 - 150. [E] Per Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281 -288. [G] D. J. H. Garling, On martingales with values in a complex Banach space, Proc. Camb. Phil. Soc. (1988). [Gl] J. Globevnik, On complex strict and uniform convexity, Proc. AMS 47 (1975), 175 - 178. [P] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326 -350. [X] Xu Quanhua, Inegalitis pour les martingales de Hardy et renomage des espaces quasi-normes, C. R. Acad. Paris 306 (1988), 601 -604.
Grothendieck type inequalities and weak Hilbert spaces
Martin Defant Marius Junge
Summary: Grothendieck's inequality characterizes operators which factor through a Hilbert space. We prove modified Grothendieck type inequalities for hilbertian operators in the sense of Pietsch. In particular, every operator from l± into a weak Hilbert space is (r,l)-summing for every 1 < r < <*> . Classical results of Kwapieh and Pelczynski can be deduced from this concept.
Introduction: In the theory of summing operators one of the most theorems is the following inequality due to Grothendieck (1956): Let n e N and A = (a^), j " and ( t ^ x n | Y. aij s i *j I
x
be a matrix such that for all sequeunces of scalars (Sj)^ x
SU
-
P
lsil
SU
P
l*jl •
Then for all Hilbert spaces (H, (o.o)) and seqeunces (Xj)^ c H and (yj)j = i n | Y. a ij ( W p I
important
-
K
G
SU
P IN!
SU
c
H
P II Xj I »
where the smallest possible constant KG is called Grothendieck constant.
In the language of linear bounded operators this reads as follows: For an operator T acting between to Banach spaces X an Y the following conditions are equivalent. 1) T factors through a Hilbert space. 2) There is a constant c>0 such that for all neJN and each matrix A: /«, -» /j || A®T: /^(X) 3) For every operator
/"(Y) ||
< c || A: / ^ - /J || .
S: /j-^X the composition TS is 1-summing.
72
Defant & Jdnge: Grothendieck type inequalities & weak Hilbert spaces
Here A®T denotes the tensor produkt of A and T . For more precise definitions see the prelimimaries below.
The aim of our paper is to apply Grothendieck's inequality to a larger class of operators than those factorizing through a Hilbert space. Hence the above conditions 2) and 3) have to be modified appropriately.
In chapter 1 we investigate the following conditions which we call Grothendieck type inequalities (theoreml.4.). For this let 1 < r < 2 < p < ° o with 1/r T: X->Y the following conditions are equivalent.
=
Vp
+
l
/i • Then for an operator
1) For every operator u: Z^X whose dual operator u* is 2-summing the composition Tu is (p,2)-summing. 2)
There is a constant c > 0 such that for all n e N and each matrix A: 7^, -> /J || A®T: /L(X) - /"(Y) ||
< c | A: / L - /" |.
3) For every operator S: l^X the compost ion TS is (r, 1 )-summing. Note that by a result of Kwapieh [KW2] condition 1) for p = l and r = 2 means that T factors through a Hilbert space (however, Kwapieh's result is basic for this paper).
In chapter 2 we study the relation of this Grothendieck type inequalities to the s o called hilbertian operators defined by Pietsch [PI2]. They are a generalization of operators factorizing through a Hilbert space and weak Hilbert operators defined by Pisier [PS2]. This concept turns out to be useful to apply the above inequalities to this larger class of operators. For 1 < r,s ^ °° an operator T: X-*Y is said to be (r,s)-hilbertian ( T e # r s (X,Y) ) if there is a constant c > 0 such that for all operators u: 12-*X whose dual operator u* is 2-summing and for all 2-summing operator v: Y -> /2 ( £ ( k1A"1/Sak (vTu) ) s ) 1 / S kelN
± c
TT2(U*)
TT2(V) ,
where ak denotes the k-th approximation number. In this case h rs (T) is the infimum taken over all constants c satisfying the inequality above. ( ( # r , h r ) = ( # r r , h r r ) , the r-hilbertian operators )
y
Ve prove that for every r-hilbertian operator T: X-+Y and for every operator S: l^X the romposition is TS (r,l)-summing.
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
73
Since for every l < r < 2 < p < o o with i/ r = l/ p + 1/2 the spaces / p and /p, (Vp + 1 /p' = 1) are r-hilbertian [PI2] we obtain the well-known result of Kwapieh [KW1] that every operator S: ll -> /p or S: /j -» /p« is (r,l)-summing.
The class of (l,<»)-hilbertian operators is just the class of weak Hilbert operators. We show that for every weak Hilbert operator T: X^Y, 1 < r < 2 the composition TS is (r, l)-summing and (,)
n rl (TS)
<
KQ - J ^ -
h^fD
every operator S: lx-+X and
|S| .
In particular, the sum operator I : *l -> /oo , ( « n ) n e N -•
( Z ock ) n € ] N k= l
is (p,q)-summing for all 1 < q < p < °° , which was proved by Kwapieh and Pelczyhski [KWP].
In this context it is natural to ask whether (*) characterizes weak Hilbert spaces (i.e. idx is a weak Hilbert operator). This we discuss in chapter 3 and we show that the interpolation space A\^ 2 °f Pisier and Xu [PSX] satisfies (•) but is not a weak Hilbert space.
Preliminaries: We use standard Banach space notations . Standard references on operator ideals is the monograph of Pietsch [PI1]. The ideal of linear bounded, finite rank operators are denoted by 2,3N respectively.
Let (A,a) be a quasi-Banach ideal. The component A*(X,Y) of the conjugate ideal (A*,a*) is the class of all operators T e 2(X,Y) such that a*(T) = sup { |tr(TS)| | S € S(Y,X), a(S) <- I } . The component A (X,Y) of the dual ideal (A ,<x ) is the class of all operators T G g(X,Y) such that T* e A(Y*,X*). We set « d(T) = «(T) .
The Lorentz sequence space (^pq , llollPq ) > 0 < P»Q - °° way. In particular,
are
deiintd in the usual
74
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
a e /lfOO
iff
a € L ,
iff
sup k a k
< °° ,
keJN
where a* = (a k ) k G N denotes the non increasing rearrangement of a e / „ . Let (e k ) k be the sequence of unit vectors in / p q (lpnq ).
For a< Banach space X the vector-valued Lorentz sequence space (/p>q(X) , |o| p > q ) consists of all sequences BX»p,q
=
x = (x^ejN c X such that
» ( IXfcl ) k € M «p.q n
For all n e N the spaces lp sequences of length n.
Let n e N , A = (^^ n
q
<
°° •
, /pJq(X) are the subspaces of /p,q , /p,q(X), respectively, of
e _,nxn K
a matrix and T e fi(X,Y). Then the operator
n
A®T : £(X) -> l q(Y) is for x = (x lc ) k= | = X ek ® xk
G
'p(X) defined by
k =ll
A®T (x) == £ A(ek) 0 Txk =
( f
kki= i
aik Txk ) "
€ lnq(Y) .
ki
For n e N the n-th approximation number of an operator T e S(X,Y) is given by an(T)
== inf { ||T-S|| |
S e 2f(X,Y) , rank(S) < n }.
For a Hilbert space H and 0 < p,q < oo the Schatten-von Neumann class CLq(H,H) consists of all operators T e g(H,H) such that Vq
)k€!N lp.q
Note that for 1 < p < ~> and 1 < p' < oo with (1)
©p(H,H)
=
© p .(H,H)
For l < q < p < ° ° a n
i / p + l/ P ' = 1
with equal norms.
operator T e g(X,Y) is said to be (p,q)-summing
( T e Il pq (X,Y) ) if there is a constant c > 0 such that for all n e N , (xj.)k" l c X
( i »Txk»" )' / p s c sup ( i ,<xk , x«>|" ) 1/q . k=l
x*eB x *
k =l
We denote by n (T) = inf c , where the infimum is taken over all c such that the above ineqality holds.
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
75
The following natural inclusion were proved by Kwapieh [KW1]. L e t l < p < r < o o , i < q < 2
( )
n p q c nr?s
and
< o o
s
with
1/q - Vp -
Vs - l/r , then
nrs < npq .
We set (II r , n r ) == (II r r , n r r ) , the r-summing operators. Recall that in this case n 2 = n 2 with equal norms. The key result of 2-summing operators is the following result of Pietsch [PI1]. An operator T € 8(X,Y) is 2-summing iff there are a probability measure n on B x * and a constant c > 0 such that for all xeX
(3)
flTxfl
< c
|<x,x*>|2
(f
dn(x*)
In this case TT2(T) = inf c , where the infimum is taken over all c satisfying the above inequality (and is even obtained as a minimum). One may suppose n to have its support in the u( X , X* ) - closure of the extreme points of B x *. A consequence is the factorization theorem of Pietsch. An operator T e 2(X,Y) is 2-summing iff there is a compact space K, a probability measure n on K, an isometry I e g( X , C(K) ) and an operator T e fi( L2(K, /*), Y ) with (S|| = 7i2(T) such that (4)
T
=
SJI ,
where J denotes the canonical mapping from C(K) into L2(K, ii) .
Note that for all nelN and for each operator v e g ( X , / 2 ) (5)
TT2(V)
<
||v*(e k )|| 2 ) V 2 .
( I k=l
A result of Tomczak-Jaegermann [TOJ] states that for n e N an operator T G 2(X,Y) with rank(T) < n only n vectors are needed to compute the 2-summing norm. More precisely, <
2 n^n)(T) ,
(6)
TT2(T)
where
n ^ T ) =- s u p { ( [
iTx^f ) l h -
|
( £ |<xk , x*>|2 )
sup
k=l
x*€Bx,
/ 2
}.
k=l
t>e a sequence of independent, normalized gaussian variables on a measure space (Q, M) . Then an operator T € 2(X,Y) is said to be 7-summing ( T e II7(X,Y) ) if there is a constant c > 0 such that for all neIN, (xk)k"j c X
1 t k=l
gk Txk 1
L2(^'X)
<- c
sup X GB
*
X*
( t k=1
|<x k , x*>| 2 ) V 2 .
76
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
We denote by TT 7 (T) = inf c , where the infimum is taken over all c such that the above inequality holds.
Note that all for nelN and an operator u e g( / 2 , X ) the 7-summing norm can be computed as follows [LPI]
Vu>
=
J Z g ki
By Pisier [PS2] a Banach space X is K-convex if there is a constant c ^ 0 such that for all n e N , v e g ( X , / 2 ) (7)
7tr(v*)
< c
n*(v) .
In this case K(X) = inf c , where the infimum is taken over all c satisfying the above inequality.
Finally, we give a reformulation of the Grothendieck inequality which will be needed later. For every n e N , e > 0 and all operators T e g(/^, , /" ) there are opertors R e g(/ 2 , / J ) , S € g ( / t , , / 2 ) such that (8)
T = RS
and
n2(S) n 2 (R*)
<
(1+e) K G
|T|
,
where K G denotes the Grothendieck constant.
1. Grothendieck type inequalities The following lemma seems to be implicitely known. However, since it is basic for what follows we give a proof. 1.1. Lemma: Let nelN and T e g(/^ o , X) . Tfien there are a sequence a e /" and an operator A e g(/" , X) such that T = AD a where
and
D a = £ ek ® a k ek k= l
||a|| 2 | A | = n 2 (T)
,
denotes the diagonal operator from /^L into l\ .
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
77
Proof: By the factorization theorem of Pietsch (3) there is a probability measure \L on K = {e k | k = l,..,n } ( the extreme points of B^/n^* ) such that for all x e f^ |Tx|
< TI 2 (T)
|<x,x*>|2
( f
dji(x*) ) 1 / 2 .
Let us define ock ••= ( M( {e k} ) )X/2 D
a
:=
n Z
e
Note that fl«|-> = f
for all
k ® a k ek
€
k = l,..,n
*K ^o , h )
, aiM
*
ii( {ek} ) = ii( K ) = 1. Furthermore we have for all (0 k ) k "i c K
k= i
m2
•
Hence | D a | = | a | 2 = 1 and
JA|| < n2(T)
and
therefore
| a | 2 | A | s TT 2(T).
Since obviously T = AD a and ( by (5) ) n2(T) < ||A|| n 2 (D a ) < | A | | a | 2 we have proved the assertion. D
From lemma 1.1. we deduce easily the following factorization for operators acting from /^o into /" . 1.2. Corollary: Let ne]N, T e g( Z^, , /J ) and e > 0 . Tfien there are sequences <x,0 e /" and an operator A e 8( / 2 , /? ) such that T where
=
D0 A D a
and
n Dp = J] e k ® /3k e k , D a k=l
| a | 2 ||A|| |j3| 2
< (1+c) KQ | T | ,
n = Y e k ® a k e k denote the diagonal
n
operators from /->
k=l
into l\ , /^o into 72 , respectively, and KQ denotes the Grothendieck constant.
Proof: By Grothendieck's inequality (8) we find operators u € g(/ 2 , /J) such that T = uv
and
n2(u*) n2(v)
< (1+e) KG
v e
g( X , / 2 )
and
||T||
Hence by lemma 1.1. there arc sequences a,|3 € l\ and operators Ax , A 2 e g( /" , / 2 )
78
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
such that
v = AJDQ, , u = DpA2 and n
-
2( v )
1312 »A2ll
>
" n 2(
Defining A = A x A 2 e g( /£ , / 2 ) yields T = D p AD a | 0 | 2 | A | Bai 2
The next lemma
*
contains
(1+e) Ko | T |
and
.
only a simple observation.
But since this is the key of this
paper ,we want to point it out. 1.3. Lemma:
Let
1 * p ^ o° , T € g ( X , Y )
, n e N , A = (aij)i,j = i
€
%( l2 » % ) '
(xj)^! c X and (y*)j = i c Y* , then we have
nth the usual modification for p = °° ) • 2) < A*®T ( £ e4 ® xj )
,
£
ej
® y* >
=
tr(vTuA)
y ® e: .
Proof: 1) A*®T ( t
e; ® xj )
=
X A*(ei) ® Tx,
i l
=
i l
^ ( I i l
a
j l
Hence assertion 1) is proved. 2)
< A*®T ( f
ej ® jq )
,
X ej ® y* >
ii
=
<
I
Cj ® £
j=i
=
=
=
Z
j=i
I
ji
i=i
aij
Tx, ,
JT
i=i
a
y < T x i > yf >
Z
ji
ii
tr( vTuA)
.
ej
® y* >
j=i
a
=
Z
ij e« ) ' v*(ej) >
But this proves assertion 2) .
Z
a
j=i i=i
ij <
=
Tu
< e i ) ' v*<ej> >
Z < T u A ( e j ) . v*(ej) ji
ij e j ) ® T x i
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
79
Now we are p r e p a r e d to prove the Grothendieck type inequalities: 1 . 4 . T h e o r e m : Let T e S(X,Y) and l < r < 2 < p < ° °
such that
l/r
= Vp + i / 2 .
Then the following are equivalent 1) There is a constant cx(l) > 0 such that for all n e N and operators A e g(/J} , / 2 ) | A®T: £(X) - /;
s
c,(T) | A: l\ -» /J | .
2) 77iere w a constant c2(T) > 0 swcfi tfwtf /or a// n e N and operators A e ^(/'L , f\) 1 A®T: £,(X) -» /"(Y) i
< c2(T) | A: £ , -> /" | .
3) There is a constant c3(T) > (? 5wc/i r/iar /or all operators S e g( /j , X ) *Yi(TS)
^
C3(T) | S | .
4) There is a constant c4(T) > 0 such that for all operators S e 2( /x , X )
IIs! • 5)
There is a constant operators u €
c 5 (T)
> 0 such that for all Banach spaces
Furthermore we have for the best constants Cl (T) < c5(T) < c4(T) < c3(T) < c2(T) < K G
Proof:
Z and for
all
Il(
1) =» 2) Let n e N , A e g( f^
Cl (T)
.
, /" ) and E > 0 . Then by corollary 1.2.
there are a,/3 e /" and B e fi(/? , /?) such that A = D|gBDa and |/3| 2 ||B|| ||oc||2
<
(1+c) KG ||A||
.
This implies || A®T: O X ) -, /rn(Y) || <
| D
8
0 i d x : /LCX)
1 B®T: ln2(X) -
Cl (T)
5
/£(
<
|a|2
|| B: /? -* l\ \
||j3| 2
i
c l( T) (1+e) K G || A: l\ -, l\ « .
Hence assertion 2) is proved. 2) =* 3) W.l.o.g. it is enough to prove assertion 3) for n e N and S e For this let (xk)k = 1 c /j A ==
I
e k ® xk
and e
g( f[ , X ).
80
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
Then lemma 1.3. yields ( I
1 TSXJ | r n
)Vr
=
| TSA(ej) | r
( [
n
)Vr
. .
<
c2(T)
[ A*: £ , -> /" 1
=
c2(T)
|S|
sup
n
sup
| S(ek) ||
=
| < X k , x*>|2 )V2
( I
c2(T)
||A| | S |
.
Hence the definition of (r,2)-summing operators completes the proof of assertion 3) . 3) •• 4) The implication 3) => 4) is obvious since by Kwapieh (2) always 4) => 5) Let Z be a Banach space and
n p 2 < 7Tr j .
u e II2( Z , X ) . By the
factorization
theorem (4) of Pietsch we find a compact set K, a probabilty measure \i on K and operators i e g( X* , C(K) ) , S e 2( L2(K, \L) , Z* ) such that u* = S J i , where J e g( C(K) , L2(K, /i) ) is the natural mapping . Furthermore we have »S H = TT2(U*) .
Since C(K)* is a Sj -Space it is simple to deduce that assumption 4) yields
( In fact, condition 4) describes an injective and maximal Banach ideal. ) Hence we get *p,2
=
np>2 ( T** u** )
=
np2 ( T** i* J* S* )
C4(T) n2(T) . 5) =* 1) Let n e N , A e fi(/2 , /?) and (xk)k" x c X . Defining
u =
I
n I ek ® xk G K— 1
, , X) assertion 1) can be deduced from lemma 1.3. and (5)
I A ®T ( £ q ® jq ) | n n
( Z
<
n
=
n
II
Tu
2(TuA*
( Z
a, Txj ||p )17P
( £ I f n
a
ii e i ) i P
)
sup
( I
x*€B/n <
c 5 (T)
n2((uA*)*)
s
c 5 (T)
I A: / " - > / "
<
||
/p
)
=
( Z
|<e k , x*>f
)
..
I TuA*(ej) 1 P
2
k=l c 5 (T) ( Z
n 2 (u*) llxki~ )
|| A: /? -•
•
)VP
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
81
2. Application to hilbcrtian operators Recall the definition of hilbertian operators. We mentioned that # x is the class of operators which factor through a Hilbert space and $i )O o is just the class of weak Hilbert operators. Although weak Hilbert spaces are rather difficult to construct there are canonical examples of hilbertian operators. 2 . 1 . Examples: i) Pietsch [PI2] proved that for l < r < 2 < p < ° ° with i / r = i/ p + i / 2 every I^p or Lp. space is r-hilbertian. We can even show that hr(Lp) = hr (L p.) = 1 ii) Using a simple extreme point argument (an observation of Geiss) it is easy to check that the Grothendieck numbers' of the sum opertor Z : /i -» /oo , (<*n)ne]N ~
a
( Z
k UlN
k=l
are bounded. But by a result stated in [DJ] this implies that Z is a weak Hilbert operator, which was first remarked in [PS1].
The crucial link between the hilbertian operators and the Grothendieck type inequalities is given by the following 2.2. Proposition: Let l < r < 2 < p < o o with i / r operator T e # r (X,Y) , n e N and
A e %{f2 , l")
|| A®T: ?2{X) -> £(Y) 1
Proof: Let n e N , x = (xk\n=1 apply lemma 1.3. we set u =
Z ek ® xk
€
k=l
= i/ p + 1/2 - For every r-hilbertian
<-
h r(T)
c B /n
S A: f2 - /? || .
and y* = Of )j = i
^( 7 2 . x )
and
e
-
v
"
=
c
Z Yk ® ek
B n
/ (Y*) '
G
k=l
2(Y,/S) •
Inequality (5) implies 7x2(u*)
=
no ( z k=l
For ( 0"1 0 = 0 )
k ® ^ )
( z k=l
i^ir )
-
In
i •
°rder
tO
82
Defant & Junge: Giothendieck type inequalities & weak Hilbert spaces
we have with v
i/ p . = i/ r , + i/ 2 E>TDPVO
=
"2(D p v o )
<-
and
n 2 ( D p ) |vofl
( ll y* »/ n ,(Y*) ) P / 2
-
and again (5)
S
( t (B y*H P V 2 ) 2 ) V 2 I v* fl j=i
<
H yfy" 1 yf
"P
*
1
-
This yields with lemma 1.4. , (1) and the definition of $ r |< A ® T ( £
ek ® Jfc )
,
I
k=l
-
| tr( DTDpvoTuA* ) |
«- ( s
fl/n,(Y*))P7r'
fly*
hr(T)
® y* >|
ej
-
| tr(vTuA*) |
j=i
<
or(DT)
MT)
w
2(DpVo)
or( DpvoTuA* ) ^2(u*) | A |
IA || .
D
As an immediate cosequence of proposition 2.2. and theorem 1.4. we get the following 2.3. Theorem: Let 1 < r < 2 . For every r-hilbertian operator T e # r (X,Y) a/irf every operator S e 9(/j , X ) f/ze composition TS « (r,l)-summing with 7T rl (TS)
<
KG
h r (T)
«S||
.
2.4. Corollary: For every weak Hilbert operator T e $1.00 (X,Y) , every operator S e fi(/j , X ) and every 1 < r < 2 the composition TS zs (rjj-summing with nr?1(TS)
<
KG - ^
h l o o (T)
Proof: First of all we note that for a positive non increasing sequence a e /x ^ and every 1 < r < °° l«lr
~
( Z
a F
k )1/r
-
( Z
k
' )1/r
l«iloo
5
-V
l«lioo
•
But this implies that every (l,°°)-hilbertian operator T € ^ j ^ (X,Y) is r-hilbertian with
hr(T)
< —I- hj
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
83
Using the examples 2.1. more classical results can be deduced from theorem 2.3. and its corollary 2.4. 2.5. Applications: i) [KW1] Let 1 < r < 2 < p * ~ and
i/ r = i/ p + i/ 2 , then
every operator S e g(/ x , / p ) or S e g(/j , / p ,) is (r, 1)-summing and
ii) [KWP] By corollary 2.4. the sum operator is (r, 1 )-summing and "r,l(D
*
10 7TT
for all 1 < r < 2 .
Moreover, by (2) r/iw implies that for all 1 * q < p < °° the sum operator is (p,q)summing and
3. Supplementary results In this chapter we want to show that the condition of corollary 2.4. does not characterize weak Hilbert spaces or operators, whereas the classical result characterizes Hilbert spaces. More precisely, we shall show that an interpolation space between the space of bounded variation and the space of bounded sequences satisfy the assertion of corollary 2.4. , but is not a weak Hilbert space.
For the following we n e e d the exact describtion of the (sequence) dual of loo\ /oo.i = { a e / ^ The
Banach
space
( /<*> j
space ( /j oo , (oDj ^
| ,
lajooj Moo^
= sup
(]T
— )
Z
) is the convexification
) . For the canonical identity i: / ^ j
a
k
([GAR])
<<*>}
of the Lorentz
sequence
-» / r , ( 1 < r <
<» ) we
have
For an abitrary Banach space X we set
/ltl(X)
-
{ X = (x k ) ke|N C X
For nelN the spaces ( / ^ j )
|
IXlL,, -
KlXkDkENll.1
and ( /„, j ) ( X ) are defined in the usual way.
The following proposition is a modified version of theorem 1.4.
< -
> •
84
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
3 . 1 . Proposition: For an operator T e fi(X,Y) the following conditions are equivalent. 1) There is a constant Cj(T) > 0 such that for all n,m e N and operators I A®T: /™(X) -» /2(Y) |
<
Cl (T)
(1+ln n) | A: f - •
/£ | .
2) 77i£/-£ « a constant c2(T) > 0 5wc/i that for all m e N a/id operators A e | A®T: £ ( X ) - ( / l f l )m(Y) | <- c2(T) | A: £
- . /f | .
3) 77iere w a constant c3(T) > 0 «IC/I r/wif for all operators S e 2( /j , X ) and fl// 1 < r < 2
nr>1(TS)
< c3(T) - I -
|S| .
4) There is a constant c4(T) > 0 such that for all neJN and operator u € Q(l^ , X ) TT2(TU)
< c4(T) (1+lnn) n 2 (u*) .
Furthermore we have for the best constants Cj(T) < c 4 (T) < 6e c3(T) s
6e c2(T)
< 12e
Proof: The implication 1) => 2) can be proved analogously to the implication 1) =* 2) of theorem 1.4. by the use of a slight modification of lemma 1.3. and the inequality 1+lnn
< 2 [
i k
k=l
For the implication 2) -» 3) note that for every 1 < r < 2 , m e N and A e fi(/^, , /" ) 1 A®T: £ < X ) -» / ^ ( Y ) || <
| A®T: /™(X) - ( / ! 5 l ) m ( Y )
^
c3(T) | A |
7
I
||
|| i d Y ® i :
(/Lfl)m(Y)
r
Therefore 2) => 3) follows from theorem 1.4. 2) =* 3) . 3) => 4) Let n e N , u G fi( / 2 , X ) . By theorem 1.4. 3) =* 5) we have for every 2 < p < oo 7Yp2(Tu)
< c3(T) 22-
n2(u*)
.
Choosing 2 < p < oo with
—t- = 3 +In n we obtain by (6) p-2 n 2 (Tu) < 2 7T^n)(Tu) < 2 nl/^l/V n p2 (Tu) P-2 2 c3(T)
n
2
P
-£.
6e c3(T) (1+lnn) 712(0*)
TT2(U*)
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
85
4) =* 1) Let m e N , ( x ^ ^ c B ^ ™ and A e fi(/^, /!)) . Then we have for
[ eekk
® xt e g(/™, X )
,
k=
by lemma 1.3. and (5)
A®T ( £ ek * x, ) I k=i n2(TuA*)
3.2.
Remark:
By
( £ lTuA*(ej)I2
-
*200
J=l
< c4(T) (1+lnn) n 2 ((uA*)*)
( |
c4(T) (1+ln n)
a standard trace duality argument one can easily check that
condition 3.1.4) means that for all u e n 2 ( / 2 , X) sup
s
-L )"J
I ak(vTu)
<
and
v €
n2( X , /2)
c4(T) n2(u*) n2(v)
.
Therefore proposition 3.1. can be considered as a Grothendieck type inequality which characterizes this sort of ( / ^ j - ) hilbertian operators.
In the following we shall closely follow the approach of Pisier and Xu [PSX] to the interpolation spaces between vp spaces. They denoted by \x the space of bounded variation Vl = i («k)k€ElN
C
'oo I
|« o l + f
l«k-<*k-ll
< °° >
k= l
Via the sum operator vl is isometric to /j . For our purpose it is enough to consider only the interpolation space A 2 = (v x , / ^ ) ij2 , 2 . Here the real interpolation method is used. We want to point out the following facts ( see [PSX] ) : (3.3.)
A 2 is a K-convex Banach space which is isomorphic to its dual.
(3.4.)
There is a constant c > 0 such that for all n e N and ( x ^ - i c A 2
( This statement is essentially a reformulation of lemma 11 in [PSX] with the use of theorem 8, see also the remark before theorem 15. )
Now we can prove the main result of this section.
86
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
3.5. Proposition: A 2 is not a weak Hilbert space, but there is a constant c ^ 0 such that for all 1 < r < 2 each operator S e fi( lx , A 2 ) is (r, 1 )-summing and nr>1(S)
s
c -p~-
||Sg .
Proof: By the definition of A 2 as an intermediate space of v1 and /«, the sum operator factors through A 2 . This implies that A 2 can not be reflexive (see [LIP] ) , But a weak Hilbert space is reflexive by [PS1]. On the other hand we note that by (3.3.) the inequality (3.4.) holds not only for A 2 but also for his dual space Aj for some universal constant c > 0 . With (6) we deduce for X e {A 2 , A 2 } , n e N and an operator u e g ( / 2 , X ) («)
n2(u) 5
2 n
-
2 sup{ ( £
»uA(ek)|2 ) 1 / 2 | | A: /£ -> f2 [ < 1 }
k= l
<- 2 ( c d + l n n ) ) 1 ^
sup{ | £ gk x,, I L2(/x,X)
k=1
^
1 2
2 ( ccl+lnn)) /
I I A: l\ -* l\ \ < 1 }
n y (u)
By trace duality this implies for every v e fi( X , / 2 ) (**) n*(v) < 2 ( c ( l + l n n ) ) 1 / 2 n|(v) = 2 ( c(l+ln n ) ) 1 / 2 n2(v) . So we deduce with (*) for A 2 , the K-convexity of A 2 and (**) for A 2 that for all and u e fi(/2 , A 2 ) TT2(U)
<
2 ( c (1 +ln n) ) 1 / 2 n r (u)
^
2 ( c (1 +ln n) ) 1 / 2 K(A2) n*(u*)
<
2 ( c ( l + l n n ) ) 1 / 2 K(A2) 2 ( c (1+ln n) ) 1 / 2 n2(u*)
=
4c K(A2) (1+lnn) n2(u*)
.
By proposition 3.1. the assertion is proved.
References: [DJ]
Defant, M. / Junge, M. : On weak (r,2)-sutnming operators and weak Hilbert spaces, (1988), Studia Math. , to appear.
[GAR]
Garling, D J . H . : A class of reflexive symmetric BK-spaces, Canad. J. Math. 21
(1969) , 602 - 608 . [KW1]
Kwapieh, S. : Some remarks on (p,q)-absolutely summing operators in lp-spaces , Studia Math. 29 (1968) , 327-337 .
Defant & Junge: Grothendieck type inequalities & weak Hilbert spaces
[KW2]
87
Kwapieh, S. : A linear topological characterization of inner product spaces, Studia
Math. 38 (1970) , 277-278 . [KWP]
Kwapieh, S. / Pelcynski, A. : The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970) , 43-68 .
[LPI]
Linde, W. / Pietsch, A. : Mapping of Gaussian cylindrical measures in Banach spaces, Theory of Prob. and its Appl. XIX (1974) , 445-460 .
[LIP]
Lindenstrauss, J. / Pelcynski, A. : Absolutely summing operators in L^-spaces and their applications, Studia Math. 29 (1968) , 275-326.
[PI1]
Pietsch, A. ..Operator ideals, Deutscher Verlag Wiss. , Berlin , 1978 and North-Holland , Amsterdam - New York - Oxford , 1980 .
[PI2]
Pietsch, A. : Eigenvalue distribution and geometry of Banach spaces, (1980) , preprint .
[PS1]
Pisier, G. : Weak Hilbert spaces, Proc. London Math. Soc. 56 (1988), 547-579.
[PS2]
Pisier, G. : Volume of convex bodies and geometry of Banach spaces, forthcomming book .
[PSX]
Pisier, G. / Xu, Q. : Random series in the real interpolation spaces between the spaces vp , Lecture Notes in Math. 1267 , 185-209 .
[TOJ]
Tomczak-Jaegermann, N. : Banach Mazur distances and finite dimensional operator ideals, Harlow, Longman and New York , Wiley 1988 .
1980 Mathematics Subject Classification ( 1985 Revision ) : 47A30 , 47A65 , 47B10 Key words : Grothendieck inequality, weak Hilbert space, absolutely summing operators.
Martin Defant Marius Junge Mathematisches Seminar Universitat Kiel Ohlshausenstr. 40-60 D-2300 Kiel 1 ( West Germany )
A weak topology characterization of t\(m) S. J . DlLWORTH1 Abstract. Let m be any cardinal. The main result characterizes l\ (m) as the only Banach lattice whose positive cone is metrizable in the weak topology. Two related theorems on ^i-sums of Banach spaces are also proved. 1.
INTRODUCTION
The three theorems of the paper concern fj-sums of Banach spaces. The first, which states that the Banach space £oo(m) contains the ^-sum of 2 m copies of itself, is essentially a translation into Banach space terms of a theorem of Pondiczery on the product topology [10]. The case m = No of this result is reminiscent of the general theorem [9] that if X is any separable Banach space containing l\, then X* contains 71/(0,1), the space of finite Borel measures on [0,1]: the connection resides in the observation that M[0,l] is linearly isomorphic to the ^-sum of c copies of Lj (0,1). This observation is used to prove Theorem 2, which says that if X is as above then X* contains 2C mutually non-isomorphic closed linear subspaces. Recall that the unit ball of a Banach space X is metrizable in the weak topology if Ar* is separable, and recall too the consequence of the Baire category theorem that if A' is infinite-dimensional then the weak topologj' on X and the weak-star topology on X* are not metrizable. However, there are still some interesting unbounded subsets of Banach spaces that are metrizable in the weak or the weak-star topology. Recall, for example, that the positive cone of the dual of C(K) (that is, the space of continuous functions on a compact metric space K), consisting of the non-negative finite Borel measures on K, is metrizable in the weak-star topology. It is not difficult, too, to see that the positive cone of £\{m) is metrizable in the weak topology for any cardinal m. Theorem 3 below the paper's main result - says that the latter characterizes l\(m) among Banach lattices. Notation and terminology are mostly standard. For every cardinal number m, i^m) denotes the space of real-valued bounded functions (xa)ae^ on a set of cardinality m with the norm sup{\ xa |: aeA}, and l\(m) denotes the space of absolutely summable functions with the norm ^2afA \ xa \. Given an indexed family of Banach spaces, {A^a : aeA], the ^i-sum, (%2acA ®^'o)b consists of all vectors x = (xa)a€A, where xaeXa, equipped with the norm \\x\\ = ^,aeA \\xa\\- The ^-sum of m copies of a Banach space X is denoted by (Em ®-^)IJ o r by (X)A ®^)i w hen A is a set of cardinality rn. As usual, No and c denote the cardinality of the natural numbers and the real numbers respectively. The positive cone X+ of a Banach lattice X is the weakly-closed cone of non-negative vectors in X. The sequence spaces £p(l < p < oo) and the Lebesgue spaces Zi(0,1) and JLOO(0, 1) of real-valued functions on [0,1] are defined in the usual wajr. Finally, the Lebesgue measure and the indicator function of a measurable set A C [0,1] are denoted by \A\ and I(A) respectively. Research supported in part by the National Science Foundation under grant number DMS 8801731.
90
Dilworth: Weak topology characterization of l\(m)
2.
RESULTS
THEOREM 1. Let m be an infinite cardinal. Then ^oo(m) contains a subspace isometric to the £i-sum of 2 m copies of £oo(m).
PROOF: ^oo(ra) has a norming set D of linear functionals (i.e. ||a;|| = sup{d(x) : deD} for all xe£oo(m)) of cardinality in. For each aeA, where A is a set of cardinality 2 m , let Da be a copy of D supplied with the discrete topology. By a theorem of Pondiczery [10] (also [5,7]) the topological product TiaeADa has a dense set {yb = (y^acA '- beB} of cardinality m. This set can be identified under the pairing (y,x) = E a c i ^ K ) with a collection of m linear functionals on ( X M ©^OO(TO))I- Let x = (xa)a€A be a vector of norm one in (J2A ®^OO("*))I and let e > 0. Select ai,a 2 ,... ,a n in A such that £ £ = 1 ||jcOfc|| > 1—e. By the definition of the product topology there exists b in B such that y2fc(«Ofc) > ||scafc || — ^ for A; = l,2,...,n. Thus (yb\x) > 1 - 3e. It follows that the mapping x -* ((z/\x)) h€B defines an isometric embedding from (Y^,A ®^OO(™))I into too{™)> REMARK: Further applications of Pondiczery's theorem to Banach spaces have been noticed. In [11, p. 236], for example, it was observed that C({0,1}2 ) (that is, the space of all continuous functions on the Cantor space {0,l} 2 ) embeds isometrically into ^oo(m) (since {0,l} 2 has a dense set of cardinality m). THEOREM 2. Let X be a separable Banach space containing £\. Then X* contains precisely 2C mutually non-isomorphic closed subspaces.
PROOF: Since X* is isometric to a subspace of ^oo(No) it has at most 2C subspaces. By the observation made in the first paragraph of the introduction, to establish the converse it suffices to show that Y = (J^ c ©Li(0, l))i contains 2c mutuall}r non-isomorphic subspaces. Recall that Li(0,l) contains a subspace isometric to £P for each 1 < p < 2 (for example, the closed linear span of a sequence of independent p-stable random variables). It follows that Y contains a subspace isometric to (^2P€A ®^p)i ^or e v e r v subset A of (1,2). The proof will be complete once it is shown that distinct subsets of (1,2) do not give rise in this way to isomorphic Banach spaces. To see that that is so, suppose that T : tq —» (^,peA (&£P)i is an isomorphic embedding, where q > 1 and A C (1>2). It has to be shown that qeA. We may assume that 6\\x\\ < \\Tx\\ < \\x\\ for some 8 > 0 and for all xetq. Since the range of T is separable we may also asume that A is countable. Let pi,p2» • • • be an enumeration of J4, let Pn be the projection from (£pi ®£P7 © • • • )i onto its first n factors, and let Qn be the projection of £q onto the linear span of the first n basis vectors of its usual basis. Since q > 1 a routine gliding hump argument proves that ||(/ — Pn)T(I — Qn)\\ —* 0 as n —> oo. Select n so that \\(I - Pn)T(I - Q n )|| < \8. Then PnT restricts to an isomorphism from (I — Qn)£q (which is isometric to £q) into tPx ® £P2 © • • • © £Pn. By standard arguments (e.g. [1, pp. 194, 205]) this implies that <7e{pi,p2> • • • >Pn}j which completes the proof. Recall the following useful concept due to Banach [1, p. 193]: two Banach spaces X and Y are said to have incomparable linear dimensions if there are no linear isomorphisms from X into Y or from Y into A'. Theorem 2 allows the following refinement. THEOREM 2*. Let X be a separable Banach space containing £\. Then X* contains 2C closed subspaces with mutually incomparable linear dimensions.
Dilworth: Weak topology characterization of l\(m)
91
PROOF: Let S be acollection of 2C subsets of (1,2) having the property that A\B and B\A are non-empty sets for all members .4 and B of S. To see that this is indeed possible, recall that the Stone-Cech compactification /?N has cardinality 2C, and that the points of /?N are in bijective correspondence with the ultrafilters of N. When regarded merely as sets of subsets of N, the ultrafilters may be put in bijective correspondence with a collection 5 of subsets of (1,2) having the required property. The proof of Theorem 2 now shows that the spaces {^p€^ ©^>)i> for AeS, have mutually incomparable linear dimensions. REMARK: Let m > c be an infinite cardinal. In [6] it is proved that there exist exactly 2 m topologically distinct compact HausdorfF spaces of weight ?n. The corresponding spaces of continuous functions have density character m (that is, they have a dense set of cardinality m), and by the Banach-Stone theorem they are mutually non-isometric. Since every Banach space of density character m is isometric to one of the 2 m quotient spaces of ^i(m), it follows that there are exactly 2 m mutually non-isometric Banach spaces of density character m. This argument does not say, however, how many mutually non-isomorphic spaces there are of a given density character. We turn now to a proof of the following theorem characterizing the Banach lattice t\(m). THEOREM 3. Let X be a Banach lattice. Then the positive cone of X is metrizable in the weak topology if and only if X is lattice isomorphic to l\(m) for some cardinal m.
Two preliminary results are required. The main idea in the first result below generalizes a fact about Hilbert space given in [8, p. 380]. I do not know a reference for Proposition 2: this result is of some interest in its own right and perhaps parts of it are known. Finally, recall that a topological space S is said to be a Frechet space [4, p. 78] if, for every A C 5 and for every x in the closure A, there exists a sequence (injjlj in A which converges to x. In particular, metrizable and first countable spaces are Frechet spaces. PROPOSITION 1. Let X be a Banach space with a normalized unconditional basis (en)^Llf and suppose that A"+ is a Frechet space is the weak topology. Then (e n )^L! is equivalent to the standard basis of t\.
PROOF: By renorming Ar, if necessary, we may assume that (en)^L1 is a normalized basis such that
for all scalars (an)^L1. First suppose that liminfn^oo | f(en) \= 0 for all /eA*, and let S = {ei + em + men : 2 < m < n < oo}. Every weak neighborhood of e\ in A'* contains a neighborhood of the form U = {xeX+ : | / f c ( z - e i ) | < 1,1 < k < N}, 1S a
collection of positive linear functionals in A'*. Let'/"— /j -f f2 4- • • • + //v. where (/fc)jkLi By assumption there exist m > 2 and n > m such that /(e m ) < | and f(en) < ^ . For 1 < k < N, it follows from the positivity of the /fc's that | //t(em +rae n ) |< /(e m +?7ien) <
92
Dilworth: Weak topology characterization of l\(m)
1, and so e\ + e m -f raenel7 0 5, whence ei lies in the weak closure of 5. Using the fact that weakly convergent sequences are norm bounded it is easy to see that e\ is not the weak limit of a sequence from 5, and it follows that X* is not a Frechet space. This contradiction implies that there exists feX* of norm one and 6 > 0 such that f(en) > 6 for all n. Then
n=l
> « ( f ) I «n I), and so (en)^L1 is equivalent to the standard basis of t\. PROPOSITION
2. (cf. (2, Prop. 4.J/J. Let P = {/ > 0 : J f{t)dt = 1} and iet
(a) P is the weak closure of Q in Lj(0,l). (b) If /„ G Q and / n -4 / weakly, then / 6 £oo(0,1). (c) P is not a Frechet space in the weak topology. PROOF: (a) Let / = X^!b=i akI{Ek) be a simple function in P, where £?i, J?2, • • • ? £n are measurable disjoint subsets of (0,1). Suppose that <7i,#2? • • • > <7m belong to Loo(0,1). For each 1 < k < n, the mapping
Xa
k I Ek |= A, and
f I{Ek)9j{t)dt) k=i
J
9j(i)dt J
for 1 < j < m. So r T( p\
r •=
/ f{*)9j(i)dt
( l < J < m ) ,
Dilworth: Weak topology characterization of l\(m)
93
and it follows that / lies in the weak closure of Q. Since the simple functions in P are norm dense in P , the desired conclusion follows, (b) If /„ —> / weakly, then the set S = {fn ' n > 1} U {/} is a weakly compact subset of Li(0,l). By the Dunford-Pettis criterion [3, p. 274] 5 is uniformly integrable. It follows that there exists M such that ||/n||oo < M for all ra, whence ||/||oo < M since /„ —> f weakly, (c) This is an immediate consequence of (a) and (b). REMARK: The converse of (b) is also true: / is a weak limit of a sequence from Q if and only if / belongs to P 0 £«,((), 1). PROOF OF THEOREM 3. It is easily verified that the weak and norm topologies coincide on the positive cone of £i(m): in particular, the weak topology is metrizable. To prove the converse, let A^ be a Banach lattice such that X+ is metrizable in the weak topology. By Proposition 1 the closed linear span of every sequence of pairwise disjoint vectors in A" is isomorphic to ti, and so by a result of Tzafriri [13. Proposition 1] A" is lattice isomorphic to L\{JJL) for some measure space (-A/,]T},/i). ^ P *s n o ^ purely atomic then £i(0,1) is lattice isomorphic to a sublattice of LI(/JL), but by Proposition 2 this would contradict the assumption that A' + is metrizable in the weak topology. So /x is purely atomic and hence X is lattice isomorphic to ^i(ni) for some cardinal in. The above arguments have in fact established the following stronger version of Theorem 3. THEOREM 3*. Let X be a Banach lattice. Then the following are equivalent: (1) (2) (3) (4)
A"*" is a Frechet space in the weak topology; A' + is metrizable in the weak topology; the weak and norm topologies on A r+ are identical; X is lattice isomorphic to t\(rn) for some cardinal m.
94
Dilworth: Weak topology characterization of l\(m)
REFERENCES
1. S. Banach, "Theorie des Operations Lineaires (Second Edition)," Chelsea, New York, 1978. 2. N. L. Carothers, S. J. Dilworth and D. A. Trautman, A theorem of Radon-Riesz type for the Lorentz space LPti .Preprint. 3. N. Dunford and J. T. Schwartz, "Linear Operators Part I," Interscience, New York, 1967. 4. R. Engelking, "General Topology," PWN-Polish Scientific Publishers, Warsaw, 1977. 5. E. Hewitt, .4 remark on density characters, Bull. Amer. Math. Soc. 52 (1946), 641-643. 6. F. W. Lozier and R. H. Marty, The number of Continua, Proc. Amer. Math. Soc. 40 (1973), 271-273. 7. E. Marzcewski, Separability et multiplication cartesienne des espaces topologiques, Fund. Math. 34 (1947), 127-143. 8. J. von Neumann, Zur algebra der funktionaloperationen und theorie der normalen operatoren, Math. Ann. 102 (1930), 370-427. 9. A. Pelczyriski, On Banach spaces containing Li(/i), Studia Math 30 (1968), 231-246. 10. E. S. Pondiczery, Power problems in abatract spaces, Duke Math. J. 11 (1944), 835-837. 11. H. P. Rosenthal, On injective Banach spaces and the spaces L00(fi) for finite measures ft, Acta Math. 124 (1970), 205-248. 12. W. Rudin, "Functional Analysis," McGraw-Hill, New York, 1973. 13. L. Tzafriri, An isomorphic characterization of Lp and CQ spaces II, Mich. Math. J. 18 (1971), 21-31.
Department of Mathematics University of South Carolina Columbia, SC 29208 U.S.A.
Singular integral operators: a martingale approach Tadeusz Figiel
Introduction. Our purpose is to sketch a new approach to proving the boundedness of a vast class of linear operators which includes, e.g., the generalized Calderon-Zygmund operators discussed in [M]. The aproach is based on estimates of operator norms which come from applying recent results concerning the Lp-boundedness of martingale transforms. In fact, the incentive for this work was the desire to extend some previously known boundedness results for operators acting in Lp-spaces of scalar-valued functions to the case of analogous spaces of X-valued Bochner measurable functions, where X is a Banach space. The recent results, due mainly to D. Burkholder and J. Bourgain, indicated that the class of the so-called t/MD-spaces may be exactly the domain in which all results concerning Calderon-Zygmund integral operators and their generalizations remain valid. (Many Banach spaces which are important in classical analysis belong to that class.) Even the simplest singular integral operator, i.e., the Hilbert transform on the real line R, has the property that its natural extension to an operator acting on Lp(Hty X) where 1 < p < oo is a bounded linear map if and only if the Banach space X is a UMD-spa.ce (cf. [Bu2] and [Bo]). In order to obtain this extension it was necessary to find such proofs which make no use of any result that does not hold in the C/MD-setting (for instance, the Fourier transform should be avoided, because it is not bounded in Z^Rj-X"), unless X is isomorphic to a Hilbert space). Thus it was a nice surprise that such austere methods could in fact lead to some results which were not less general than their counterparts established earlier with no restrictions on the range of admissible methods. Our approach is indirect in the following sense. Rather than trying to prove that some "classical" operators are bounded, we started from considering certain rather new operators (cf. [F]), which in our opinion have a basic nature. (All the "singularities" which can occur in our context are neatly packaged inside the basic operators.) Having established precise estimates for the norms of those basic operators, we can take up the "general case". We just look at the class of those operators which can be realized as the sum of an absolutely convergent (in the operator norm) operator series whose summands are simple compositions of our basic operators. Then it turns out that the choice was sufficiently efficient for that class to contain so-called generalized Calderon-Zygmund operators (cf. [DJ], [M]) and much more. We illustrate the latter fact in the case of the famous T(l)-
96
Figuel: Singular integral operators: a martingale approach
theorem of David-Journe. In that case our aproach is applicable to kernels which (instead of being Holder continuous off the diagonal) only have locally bounded mean oscillation which is dominated by an apropriate function of the distance to the diagonal of R d x R d . This short account cannot give full justice to all the important work done earlier in this area. The reader is advised to look into [M] to see some things which are not mentioned here and which could have influenced the present author. We cannot adequately cover here all possibilities given by this aproach. Most of this exposition is devoted to stating the appropriate definitions and sketching the main steps. The abstract scheme. Let us present first in abstract terms the general setting in which we shall be working. Fix d > 1 and let B denote the space of bounded linear operators from the Hilbert space L2(Rd) into itself. We shall consider a triple (CZ, WB, T), where WB and CZ are linear spaces with CZ C WB, while T : B —> WB is a linear injection. We want to characterize those operators T £ B such that T(T) £ CZ and we shall do this entirely in terms of the class CZ. Thus we shall produce a condition, say C, which is formulated in terms of CZ only, so that, if J E CZ, then J satisfies C iff J = T(T) for some bounded operator T 6 B. Let us write CZO = T(B) 0 CZ. The main advantage of our introducing the space CZ stems from a rather simple and transparent structure of that space. This is what makes CZ a good intermediary between the class of L2-bounded operators and the class, denoted below by CZK, of appropriate singular kernels. (The class CZK is generated from CZ by means of a linear map \I> which will be specified later on.) However, there may be something more intrinsic involved in this construction. In fact (besides the models based on the Haar system which we are going to describe in this exposition) there are other natural models for the triple (CZ, WB, T) and the just mentioned map #, which share a very similar structure. In those models (which are defined in terms of other natural systems of functions, like the spline bases (cf. [CF]) or the wavelets (cf. [M]), the maps T and ^ are replaced by their relatives, but the linear spaces WB, CZ and CZO remain essentially the same. The main result in this exposition is a relative of the T(l) theorem of David-Journe, because for those triples which we define below the condition C turns out to be analogous to the condition that T(l), T*(l) are both in BMO, where T = Tf-^J). Namely, our condition C says that certain two objects (defined in terms of J only) should belong to a linear space closely related to BMO. Actually, for each J G CZO the operator T = T - 1 ( J ) will be associated in the usual sense with the kernel #(J) defined on ft = Kd X R d \ {(x, x) : x G R d }, where # will be a natural linear map \I> : CZ —> K, (and K, stands for the set of all measurable "kernels" K(x,y) on ft). (In the case of Haar-type systems the resulting kernels are absolutely integrable on every diadic cube / C R d x R d whose interior is a subset of ft.) It should be mentioned that the set CZK — V(CZ) of all kernels covered by our result will be rather rich. The following remark is meant to clarify the relationship between our class CZK of all kernels and its subset ty(CZO) consisting of those kernels that are associated with some members of the set T~1(CZO) of our generalized Calderon-Zygmund operators.
Figuel: Singular integral operators: a martingale approach
97
In our models the kernel, i.e., the null space ^ 1 (0), of the map \fr is always a subset of CZO. More precisely, for J G CZ the property # ( J ) = 0 is equivalent to J = T(Afy) for some <£ G I>oo(Rd). (The operator M 0 G £ is defined by M 0 (/) = 0 / for / 6 L2(Rd).) Thus, the kernel K(x,y) (i.e., an element of CZK) cannot determine a unique operator T G B such that K = ^(T(T)), however, it carries enough information in order to find out whether or not it corresponds to such a T. Namely, if K G ^(CZO) and J\ G CZ, then ty(Ji) = K implies Jx G CZO, i.e., the answer does not depend on making a particular choice of Jx in ^~X(K). Actually, the line of our argument has many points in common with the approach used in [DJ] and especially in [M]. The fact that the differences are not merely superficial becomes clear when one considers the problem of the boundedness of the Calderon-Zygmund operators in the Lp-spaces of X-valued Bochner measurable functions. The earlier approach, based e.g., on Cotlar's lemma, works only if X is isomorphic to a Hilbert space. However, the work of D. Burkholder and J. Bourgain has shown that in this context the proper class of Banach spaces is that of CTMD-spaces. On the one hand, it is known (cf. [Bu]) that Calderon-Zygmund singular integral operators are bounded in the vector-valued UMD case, on the other if the Hilbert transform H is a bounded operator in Lp(X) for some value of p, then the Banach space X must be a UMD space [Bo]. The estimates on which our proof is based depend only on martingale transform techniques and hence our results are applicable to the spaces of Jf-valued functions whenever X is a UMD space. For the sake of simplicity we shall skip the information about the X-valued versions in most statements given below. The Haar system in Rd and related structures. The orthonormal system of Haar functions has been first introduced on the unit interval [0,1]. Later the analogues of that system have been developped in the multi-dimensional case (cf. [C]). In many applications, including the present context, it is not good enough to take tensor products of Haar functions on R - the supports of the functions in our systems on R d should look like cubes rather than like parallelepipeds. Below we recall the definitions and introduce a notation which will be convenient for our purposes. First let d = 1. If I C R is an interval, then by |/| we denote its length. By Z or Z 1 we denote the set of integers. Diadic intervals in R 1 are sets of the form / = [A;2n, (k + l)2 n ), where k, n G Z. We let 1(1) denote the set of all diadic intervals in R 1 . It will be important for us that the group Z 1 acts in a natural way on X(l). Given m G Z, every interval J G 2"(1) is moved by m units of its own size, i.e., the image of I is the diadic interval / + m\I\ = [inf / + m|/|,supI + m\I\). The Haar functions on R 1 are indexed by the set J(l), i.e., HAAR(\) = {Xl : I e J(l)} (and each Xj is supported on the set I). All of them can be in a simple way generated in terms of one of them. Namely, Xro i) = ho i) ~ If1 i) an( ^ ^ or ^ ^ -^(1) o n e ^ ias
)'
98
Figuel: Singular integral operators: a martingale approach
The system HAAR(\) is a complete orthonormal set in L 2 (R). Let us put for I G
x)=xi,
x? = m"* i/ = ix/i-
Now let d G Z be greater than 1. Diadic cubes in R d are sets of the form I = (Io + rn |/ 0 |) x (Jo + n2|Jo|) x ... x (Jo + n d |J 0 |) where n = (ni, n 2 , . . . , n^) G Z d and Jo G X(l) is a diadic interval (which can be taken to be [0,2n) for some n G Z). We let J(d) denote the set of all diadic cubes in R d . Given I = h x . . . x Id G J(d) and e = (e\,..., £<*) G {0, l} d , we put
Clearly, the function Xj is supported on the set I. Let us we put
The Haar system on R d is defined by the formula HAAR(d) ^{x^'.Ie
l(d),e G NS(d)}.
It is a complete orthonormal set in ( ) Again, it is important that the group Zd acts in a natural way on the set Z(d). Namely, given m = ( m i , . . . , md) G Z d , every diadic cube I G Z(d), say I = I\ x . . . x Jj, is transformed by m so that the image of I is the diadic cube (Ii+mi\Ii\)x.. .x(Id +rarf|Jd|), which we again denote by J-fm|I\ (here \I\ refers to the size of I rather than to its volume). Now we shall explain the notation which we use when dealing with functions on the space R 2d regarded as the product R d x R d (a pair (x, y) G R d x Rrf being identified with the point (xu ... ,xd,yu ... ,yd) G R 2d ). We shall often represent elements a G NS(2d) as pairs (e,r/), where e, 77 G {0,1 } d . Note that every element /i = X/ of Ifi4AR(2 (8) /i(y> (i.e., /i(x,y) = hlx\x)hM(y) for .each (x,y) G Kd x R d ). Indeed, it is clear that we can take /&(*) = Xjt and h^ = X/2J where J1? I2 G X(c?) are apropriate cubes and not both e and 77 are equal'to 0 = (0,..., 0). Finally, we can specify the space WB of weakly bounded elements. We shall take WB = loo(HAAR(2d)), the space of bounded functions on the countable set HAAR(2d) equipped with the usual || - ||oo norm. The map T : B —> WB from the space of bounded linear operators on L2(Rd) into WB is denned by the formula
in which (•, •) stands for the natural inner product on Before we describe our subspace CZ C WBy let us mention that it will contain all finitely supported elements of WB. Every such element, interpreted as the corresponding
Figuel: Singular integral operators: a martingale approach
99
linear combination of elements of HAAR(2d), obviously determines a "kernel" on R 2d . The map # will be the natural extension of the latter correspondence. Our further work with the kernels of operators in terms of HAAR(2d) will be made more transparent by passing to a different parametrization of that set. Consider the set IS(d) = I(d) x Zd x NS(2d) as a new set of indices. For i G IS(d), say i = (J, m, (e, 77)), let hi = X/+m|/i ® Xj- Obviously, hi coincides with an element of HAAR(2d), namely fy/,m,(c,i7)) = X(/!fm It is clear that the map i —> hi establishes a one-to-one correspondence between the index set IS(d) and the Haar system HAAR(2d). The latter map induces an isomorphism between WB and the space loo(IS(d)). Formally, to each J G WB there corresponds the element J G loo(IS(d)), defined by J(J,m,e,7/) = J(^(/,m,(c,T?))). Thus loo^IS^d)) is just another incarnation of WB. We often prefer to work with J rather then with J, we hope that it will not do much harm if we sometimes skip the " ~" in our notation. The choice of the subspace which will be our CZ is not canonical. Knowing how the proof goes and which estimates of the basic operators can be improved (perhaps by taking advantage of some special features of the case that one is interested in), one can modify the choice of CZ so as to obtain a somewhat bigger subspace for which the arguments still work. At this moment it is preferable to choose a simple variant. For n G Z d , we let Sn denote the indicator function of the subset of IS(d) consisting of those (/, m, (e, 77)) such that in = n. Later on we shall specify a real function w on Zd with u>(n) > 1, in terms of which we define the space CZ and the norm ||*||cz> u s i n g ^ ne formula \\J\\CZ
= E "WH*" J~n°o •
nez* the space CZ will, of course, consist of those J G WB such that || JHc-^ < 00. In fact, some properties of CZ are true under the minimal assumption inf u;(Zd) > 1. For n G Z d , we let |n| = H n ^ = max; \m\. Finally, let us recall a definition of the spaces H{iad(Kd) and BMOdiad(Rd). (Both definitions work also in the case of X-valued functions, where X is a Banach space.) A function / G Loo(R.d) is said to be a diadic atom, if there is a diadic cube I G Z(d) such that ljf = / , H/H^ < l i p 1 and (/,1) = 0. The space Hfiad(Rd) consists of those functions / G Li(R.d) which can be represented as ^ , *»/i, where the /,-'s are diadic atoms and Y^i \t%\ < °°- Then ||/||/f<*.-a<*(Rd) is the infimum of all those sums YLi 1**1 < °°A locally integrable X-valued function g on R d can be integrated against any diadic atom / on R d . It is easy to verify that sup{ / fgdxifisa. Jn*
diadic atom on Rrf } = sup \I\~*c inf 111 Ag — c)\\ . iei(d)
100
Figuel: Singular integral operators: a martingale approach
Those functions g for which the latter expression is finite are said to have bounded mean oscillation on diadic cubes. The value of above expression is denoted by ||<7||£MO"<"I(R')The space BMOdtad(Tl.d) consists of all those functions (modulo the constant functions on Kd). It is well known that the space BMOdiad(Rd) can be naturally identified with the dual space of Hdtad(TLd) in the way we have described (say, in the scalar case). Let us recall a characterization of elements of BMOdtad(R.d) in terms of their coefficients with respect to HAAR(d) (in the case of scalar-valued functions). We put, for a scalar valued function a defined on the set T(d) x NS(d), 1
E K^
E
J£X(d),JCI
e£NS(d)
The condition ||a||6mo/Rd\ < oo is necessary and sufficient for a to be the family of Haar coefficients of a function g £ BMOdtad(Rd). Let us mention that the latter statement should be formulated in a different way, if one considers the X-valued case. We let bmo(Rd) denote the space of those a such that |M|&mo(R
Then to obtain the second formula we just apply the first one to T*
£
= III* E
jex(d),\j\=\i\
jex(d),\j\=\i\
Observe that in both formulae each term on the right-hand side is well defined but the infinite sums are in general case divergent. Since those terms make sense even if TT is replaced by any element of WB, and since elements of CZ are so nice, it is not unnatural to introduce two linear operators, say 0 X and Oy , which map the space CZ into scalar functions on the set HAAR(d). We simply put, for each (J,e) G I(d) x NS(d), jei(d),\j\=\i\
* E jei(d),\j\=\i\
Figuel: Singular integral operators: a martingale approach
101
Since u;(n) > 1, the above sums are absolutely convergent for each J G CZ. In fact, one has |0z(J)(X/)l < M1*||^|lcz f o r z e ix>y} QIid f o r e a c h ( J » c ) € Ad) x NS(d). Now we can state the main result. Theorem. Suppose that the sequence LJ in the definition of CZ satisfies the estimate w(n) > clog(2 + |n|) for some c> 0 and every n G Z d . Then, for J G CZ, J G T(B)
if and only if
0X( J), 0 y ( J) G 6mo(Rd).
Moreover, if this is the case (i.e., if J G CZO), then the operators T = T~ 1 (J) and T* are both bounded from Hfiad(Rd) to Li(Rd). Consequently, T and T* are bounded as maps from £oo(Rd) into BMOdiad(Rd) and hence from Lp(Rd) into Lp(Rd) for 1 < p < oo. .Furthermore, if T(T) G CZO and X is a Banach space with the UMD-property, then the statements made above concerning the boundedness of T in various function spaces carry over to the case of the spaces of X-valued Bochner measurable functions on Rd. Remark 1. Observe that the theorem implies in particular that, if T G B satisfies J = T(T) G CZ, then T has all the boundedness properties listed in the theorem (there is no need to verify that Tl, T*l are BMOdtad-functions, because this is assured by the theorem). Remark 2. The BMOdtad-condition is weaker than the classical BMO-condition, hence our assertion concerning the Lp-boundedness covers a wider range of operators in B. The elements of that class (i.e., of the set 1~1{CZD)) usually are not L^-BMO and Hi-Li bounded, since the latter implies a stronger property, namely that both T(l) and T*(l) (which, as we know, are well-defined elements of BMOdiad(Rd)) belong to BMO(Rd). Now, for a properly chosen subclass CZreg of CZ (to be specified below), our main result implies easily that the condition T(l), T*(l) G BM0(R.d) is necessary and sufficient for the Loo-BMO and #i-Li boundedness. (The trick is that a function g on Kd is in BM0{Kd) iff the translates {gz : z G 5} form a bounded subset of BMOdiad(Rd) for a sufficiently big subset S G R d (we let gz(x) = g(x — z) for x,z G R d ).) The relevant property of the set S is that for every cube Q G R d there is z G S and a diadic cube / G Z(d) such that Q + z C / and the ratio of volumes of / and Q is bounded.) In our setting, the class CZreg can be chosen to consist of those T G CZO whose all translates Tz by elements of a subset S of R d have their images in WB, contained in a bounded subset of CZ, i.e., sup z e 5 ||^(^z)|lcz ^ °°* ^ good choice for S is, for instance,
LUz U t } "
The above criterion for L^-BMO and H\-L\ boundedness of operators can be significantly improved, yet the proper setting for such results seems to be in the context of other systems of functions on Kd which we have already mentioned. The corresponding results for those systems can be deduced from the results presented in this exposition with the use of the equivalence established in [F]. Application to the Hilbert transform. Now let us see how this result can be applied in the simplest non-trivial case of a Calderon-Zygmund integral operator, i.e., to the Hilbert transform H. A thorough investigation of this special case will shed some light
102
Figuel: Singular integral operators: a martingale approach
on typical problems arising when one studies such operators. So let us pretend, for the time being, that we do not know that H is L2 -bounded (or that we want to obtain, using our main result, the known fact that H is bounded in a space Lp(H, X) of vector-valued functions, where 1 < p < oo and X is a l/MD-space). We want to construct a bounded operator on ./^(R), s a v 2 \ which is associated with the kernel K defined as K(x, y) = ^~ for x, y G R, x ^ y. As we know, such an operator (if it exists) is not unique, but our T should also be anti-symmetric (which means that (Tf,g) — -(f,Tg) for / , g G L2(Rd)) and real (i.e, it should take real-valued functions into real-valued functions). We first find a suitable element of CZ, so that we can apply of the results we stated above. To do this we consider the expressions ct(J, n), where / G 2"(1) is a diadic interval and n is an integer, defined (if n ^ 0) as
a(I,n) = / K(x,y)dxdy, J(I+n\I\)xI because the £L4j4i2(2)-coefficients of K can be easily expressed as soon as we know the a(I, n)'s. Namely, if I', I" G 1(1) satisfy / = /' U I" and I" = I' + |/'|, then for n ^ 0
Of course, for n = 0 the above definition of a(/, n) would not make much sense, hence for the time being let us just take a(/, 0) to be 0 for each / G ^(1) and examine the consequences. Reasons for making that particular choice are made clear later on. If n j£ 0, then the a(7, n)'s are well-defined, because K is absolutely integrable on the set (7 + n|/|) x / . We could compute the numerical value of each those a(7, n)'s exactly, but now it will be enough for us to observe that a(7, n) = b(n) • |/| for n ^ 0, where the sequence b(n) satisfies b(—n) = 6(n), and b(n) = n" 1 + O(n~2) as n tends to oo. Because of our earlier choice, letting 6(0) = 0 we have now a(I,n) = b(n) • |/| for all n G Z. Using the latter relation we obtain simple formulas for all / G X(l) and n G Z \ {0} - 1) - 6(2n + 1)), + 1)) • Now we define J G WB by letting for h G HAAR(2), say h = X(/+ n |/j)x/ is now a legitimate choice,
wnere
^= 0
J(&) = \ ((1 + (-ir+")6(2n) + (-l)<6(2n + 1) + (-l)"6(2n - 1)) . Observe that J G CZ, because one has |^(X(/+ n |/|) X /)l = O(n~2) as |n| tends to oo. It is very easy to check that our J satisfies 0 X ( J ) = ® y (J) = 0, thus by our main result we have shown that J G CZO, i.e. J = T(T) for some T G B. It is also easy to
Figuel: Singular integral operators: a martingale approach
103
check that the operator T is anti-symmetric and real, as we requested (we started with a real-valued anti-symmetric kernel). Now, if we just tried to choose any real values for the a(/, 0)'s appearing in our recipe for J, so as to obtain an anti-symmetric element, say J\ G WB, then for each I G ^T(l) there would be two equations to be satisfied, namely Ji(Xj ® Xj) = 0 a n d Ji(Xj ® Xj) = —Ji(Xi ® X/)- Using the formulae obtained earlier we can conclude that a(J',0) = a(J",0) = 0. Thus our choice of the a(/,0)'s is that only one which yields a real-valued and anti-symmetric J. It should be quite clear that, if one knows beforehand about H that it is an L^bounded, anti-symmetric, real operator which corresponds to the kernel cK, where c = — ^, and hence its iL4AR(2)-coefficients (i.e., the element T(H) G WB) coincide with those of cJ, then it now follows that T = ^ff, because T is a one-to-one mapping. On the other hand, if we insist on not using anything from the classical theory of H (which is necessary if we want to obtain a new proof), then we still have to find the "kernel" of T, i.e., the element \&(Y(T)) = ^ ( J ) , and show that it is equal almost everywhere on R 2 to the function K(x, y) with which we started. We also should establish the formula
(Tf,g) = J K(x,y)f(y)g(x)dydx, say for those / , g G L 2 (R) which are supported on non-overlapping diadic intervals. To do this we consider three sets of functions. We let Sa = HAAR(2) \ {X/xj : / G X(l),(e,?7) G NS(2)}. Then let Sb be the set of the indicator functions of those Q G 2"(2) such that Q is on one side of the diagonal, and no Qf stricly containing Q has that property. Finally, let Sc consist of the indicator functions of those Q G 2"(2) which are either of the form [0,2 n ) x [-2 n , 0) or of the form [-2 n ,0) x [0,2 n ). We check easily that for / G Sa U5& U 5 C , one has J #(J)f = / Kf. Of course, this is enough to show that the two functions are equal almost everywhere. The integral formula for T(/, g) is then an easy consequence. Comparison of kernels in CZK with standard kernels. A typical class of generalized Calderon-Zygmund operators is that which corresponds to the so called standard kernels (cf. [DJ], [M]). In the case of the d-dimensional space R d , where d > 1, the latter class consists of those functions K defined on the set Q = R d x R d \ {(a:, x) : x e R d } such that, for some S > 0 and B < oo, one has \K(x,y)\
\K(x, y) - K{x, y')\ < B\y - y'\ \x -
y\'*-\ y\-d~\
for all (x,y), (x',y), (s,y') G Q such that \x - x'\ < \\x - y\ and \y - y'\ < \\x - y\. Given a standard kernel K, does it belong to the class CZK ? The answer is yes, if K is associated to a bounded operator T on L2(R d ) (which is then said to be a generalized Calderon-Zygmund operator). In that case we expect K
104
Figuel: Singular integral operators: a martingale approach
to coincide with #( J) where J = T(T). Of course, J G WB, and we should check that J G CZ to make sure that #( J) is defined, showing at the same time that #( J) G CZK. (Then it is easy to verify that #( J) = K almost everywhere on R 2d , and hence K G CZK.) It will suffice if we get an estimate of the form
for all I G T(d), (£>*?) € NS{2d) and all n G Z d such that |n| is sufficiently big. say > n 0 . (The subset of CZ determined by the above condition will be denoted briefly by CZs.) Such an estimate follows rather easily from the second and third condition in the definition of standard kernels. Thus the class CZK described in our main result contains the kernels of all generalized Calderon-Zygmund operators in the sense of [DJ] or [M]. The answer may sometimes be negative (consider the kernel K(x,y) = \x — y\~d on 0, the case of dimension d = 1 being rather easy). On the other hand, in case of a general standard kernel K(x, y), the question is harder. In order to construct a J G CZ such that *&(J) = K we should determine all the HAAR(2d)-coef&cients of J, i.e., the values «/(X(/+n|J|)xl) for I e Z"(<0> n G Z d and (£,77) G NS(2d), in a manner consistent with the values obtained by integrating K against the Haar functions. Those coefficients whose index (/, n, e, 77) satisfies n ^ 0 are defined uniquely by K and their absolute values are dominated by ci?|n| . (We know this already for |n| > no, and if 1 < |n| < no, this estimate follows from the first condition on the standard kernel.) Thus a suitable J (with a finite CZ-norm) can be found for our K if and only if the remaining values of J (those taken on the remainder of the set HAAR(2d)) can be chosen to be bounded and so that \I>( J) — K. Those remaining values correspond to the diagonal elements of HAAR(2d), i.e., they form the element 60 J, which is not determined uniquely by ^ ( J ) , hence a fortiori by K. However, the condition &(J) = K is in this way reduced to a countable system of linear equations. (There is one equation for each diadic cube which is maximal in the family of all diadic cubes whose interior is a subset of Q, and there are also equations corresponding to diadic cubes which are located as those in the definition of the family Sc.) Finding a bounded solution of that system is the heart of the matter. We are going to say more on this subject in another paper. Elements of our class CZK need not be Holder continuous off the diagonal. Their definition given in terms of CZ is at the same time a description in the sense of constructive theory of real functions. Here we shall describe only the subclasses of CZK that are induced by the classes CZ$, which generalize the standard kernels. Recall that, if S > 0 then CZS = {J eCZ : |J(/,n,e,r/)| = O(\n\-d'6)}. In view of the obvious estimate \J(XQV)\ < \\XQV\\H^d{Q)nQnJ)\\BMOdiad(Qr
valid for every Haar function
€ x
d
that Q = (I + n|/|) x /, / G I(d), n G Z |n| > 1, the class CZ6 contains the set {JeCZ:
It turns out that the latter set equals CZ& (this can be checked easily using the characterization of BMOdiad(Q), in terms of the JMA,R(2d)-coefricients), i.e., for K G V(CZ) the
Figuel: Singular integral operators: a martingale approach
105
condition K G ^(CZs) is equivalent to the following estimate
Hl(/ +n |/|)x^ll SMO .,. ((/+n|J|)x/) < 5|»r'-*m- d ,
I e Ad), „ € Z', |n| > 1,
which should be compared with the second and third conditions on the standard kernels. Basic operators and the decomposition of CZO. operators in Ir2(Rrf).
Let us present our basic
Lemma 1. For each n G Zd there are operators Tn and Un G B such that for each Xj G HAAR(d) one has Tn(X/) = X/+n|I|>
^nCX/) = X?+n|/| - Xj-
Moreover, the operators Tn and Un are both bounded from Hdtad(R.d) to Li(R d ) and from Loo(Kd) into BMOdiad(Kd) and hence from Lp(Rd) into Lp(Kd) for 1 < p < oo. If X is a Banach space with the UMD-property, then the operators Tn and Un are both bounded in the corresponding spaces of X-valued Bochner measurable functions on R d . In each of those cases the norms ofTn and Un are < Clog(2 + |n|), where C < oo depends only on p and X. This lemma essentially appears in [F], where it is proved in terms of the Haar system on the interval [0,1] (cf. Theorem 1) and it is explained how to deduce the analogous fact for the system HAAR(1) using a simple rescaling procedure. It is observed in [F] that there is no basic difficulty in extending the martingale method used to prove Theorem 1 to the case of R d and we shall do that in detail elsewhere. In the next lemma we introduce the operators (namely Pa and P*) which are the diadic version of the operators known as paraproducts (cf. [DJ], [M]). Lemma 2. For each a G bmo(Rd) there is Pa G B such that for each %/ G HAAR(d) one has Po(X£/) = a(J, e )|7|- 1 l / . Moreover, the operators Pa and P* are both bounded in the same spaces as the operators Tn and Un from Lemma 1 and in each of those cases the norm of Pa and of P* is < C\\9\\bmo(iLd)> wnere C < oo depends only on p and X. In the case of spaces of scalar-valued functions Lemma 2 follows from standard facts (we give a proof in the final section — the case where d = 1 contains all the ingredients needed in the general case). The proof in the case of X-valued functions, which relies on an estimate due to Jean Bourgain (October 1987, unpublished) and is more technical, will be presented elsewhere. We start with more notation. (The reader who is not familiar with bases in function spaces on R d is advised to think first of the case of dimension d = 1.) Recall that the set IS(d) = l(d) x Zd x NS(2d) can be regarded as a convenient set of indices for the system HAAR(2d) and that there is a natural isomorphism between WB and loo(IS(d)).
106
Figuel: Singular integral operators: a martingale approach
We need more {0, l}-valued functions on IS(d). The three new ones will be denoted by 8NN, 8ZN and 8NZ. They are the indicator functions of the following three subsets of IS(d): {(J,n,e,i/) : e £ 0, rj ^ 0}, {(/,n,e,iy) : e = 0} and {(J, 11,6,17) : ?7 = 0}. We shall show that the elements of the space CZ admit a nice decomposition. Each J e CZ can be written in the form J = Jo -f J far , where Jfar £ CZO and J o satisfies S0J = J, 8NN J = 0. (Let us remark that JQ contains the same information as the pair ( 0 x ( J ) , 0 I ( J ) ) which appears in the main result and this shows a link with Lemma 2.) The element Jfar is constructed below in terms of J, from the operators appearing in Lemma 1 and other operators which are diagonal with respect to HAAR(d). For elements J G CZ we shall use the notation ||J|| op which will denote either +00 or the norm of T~1(J) in the sense of the space of operators which is being considered at the moment. (We prove first that some expressions define bounded operators on L2 and later want to use the same formulae again in order to estimate, for instance, the norm from Loo(Rd) to BMOdiad(Rd).) The component 8NN Jfar is defined as the sum of the series SNN = 5^nGZd We shall estimate the ||*||op of the terms of SNN and show that their sum is < (This argument utilizes the operators Tn from Lemma 1.) It will follow that the element SNN J, which is the sum of that series, belongs to CZO and is bounded in all the norms which we consider. Then a similar trick can be done in order to express the component 8ZN Jfar as the sum of a series of the form SZN = £ n€Z -\ {0 >(M ZJV 'J ~ J z N ) . (We let the expressions JZN stand for suitable compensating summands (defined below) which satisfy 8Q8ZN JZN = JZN. Hence the sum of the series SZN vanishes where 8ZN = 0 and it mimics J where 80 = 0 and 8ZN = 1.) This part uses the operators Un from lemma 1. The third component of Jfar, i.e., 8NZ JfaT, is obtained in an analogous way as 8ZN Jfar. We use a series of the form SNZ = J2nezd\[o}(^NZ J - JnZ), etc. (just swap the Z's with the TV's). This case is dual to the previous one, the relevant operators being now the
U?s. A fundamental estimate is the following simple inequality which reflects the unconditionally of the Haar system on R d in various function spaces. This inequality is valid for J G WB (with ||-||op denoting the norm in the sense of the space of operators acting either on Lp(Kd) with 1 < p < 00, or on BMOdiad(Rd), or Hfiad(Rd), or the X-valued version of one of the above, X being a UMD-spa.ce)
\\S06NNJ\\op < C P o ^ J I L , and the constant C < 00 may depend on the operator norm which is being considered, but it does not depend on J. For instance, if ||-|| denotes the norm of operators on L2, then we can take C = D2, where D = 2d — 1. (Being more careful, one can get C = D (rather than D2) in the last estimate, the case of d = 1 is really simpler.) The latter statement is quite obvious, because the operator T can be decomposed into D2 components (corresponding to fixing the values of e and 77 in the indices of J). Those components either are diagonal operators with respect to the complete orthonormal system HAAR(d) in L2(Rd) (if £ = rj) or they can be written as compositions of such diagonal
Figuel: Singular integral operators: a martingale approach
107
operators with certain permutations of HAAR(d) (if e ^ ?/), hence their operator norms in L 2 (R d ) are dominated by the supremum of the coefficients. Thus T = T~1(808NN J) G B. A similar argument works for all the operator norms which are listed above, because the unconditionally of the Haar system in all those norms is known to be equivalent to X being a UMD-spa.ce (and the 1-dimensional Banach space R is a UMD-spa.ce). Now, using the operators T n , we can reduce the estimates for 8n8NNJ to the fundamental estimate. It will be convenient for us to have a notation for the action of the group Zd on the space WB (regarded as the space /oo(JS(d))). We put for n G Zd and J G WB (Tn( J))(/, m, e, 77) = J ( / , m - n,e, rj). For J G WB we have 8n8NN J = T(T n o T - 1 ( T _ n ( ^ n ^ N J))), hence we obtain It *T cNN Til
\\8n8
^ \\rr\ || \\rr
(c
J|| op < ||T n || \\T-n(8n8
cNN T\||
^ /^WT II II JT cNN T||
J)|| o p < C\\Tn\\ \\8n6
J]]^.
It should be clear now how to complete the estimate of ||£NN«/far||Op m ^ ^ e cases stated in the Theorem. (Here and also below the case of Loo-BMOdiad and of H(iad-Lx estimates may require some additional routine considerations which are better left to the diligent reader.) Now we take up the second term, i.e., 8ZN^ Jfar. We have to describe those compensating summands which were denoted by JZN in the series for 8ZN Jfax. We simply take JZN = T-n(8n8ZN J). Consider the diagonal operator with respect to HAAR(d), say A, defined by the formula A(xJ) = «/(/, n, 0, vDx] ^^^ observe that T(Un o A) = SnSZN J - T-n(6a6ZN
J).
This allows us to show that the ||-|| op norm of the difference on the right-hand side is
which is again sufficient to complete the estimate. The third component of Jfar is handled in an analogous way. Having constructed the element Jfar G CZO, one can evaluate the difference Jo = J — JVar, and note that it is closely related to our formulas for Oy(J) and 0 X ( J). Namely, if those two elements are both in 6mo(Rd), then Lemma 2 yields two operators Pa and Pa' G B so that a = 0 y ( J), a' = 0 X ( J) and T(P a ) = 6o6ZNJo,
T(P;,) = SoSNZJo.
Since J o = $o$ZN Jo -f 8o8NZJoy one obtains that J o , and hence also J, belongs to CZO. The converse is true too, i.e., one can show that, if T G B satisfies the relation T(T) = J = 6QJ, then both elements 6ZNJ and SNZJ are in T(H) and hence 6ZNJ = T(P fl ) and 8NZJ = T ( P J ) , where a, a' G 6mo(Rd), a = 0 y ( J ) and a' = 0 X (J). This completes our sketch of the proof of the main result.
108
Figuel: Singular integral operators: a martingale approach
The boundedness of paraproduct operators. For k G Z let Ek = Esfc denote the operation defined in the space of locally integrable functions on R, which transforms a function / into its conditional expectation with respect to the a-field S* generated by the diadic intervals of length 2~k. We let A* = Ek — Ek-\> Consider the bilinear operation P defined by the following formula
where /,# are locally integrable functions on R. (Throughout the argument one may assume, to keep the things simple, that g is in the linear span of HAAR(1). The estimates proved for that special case would suffice for our purposes.) For 1 < p < oo and for suitable functions gy we would like to obtain an estimate of the form
\\P(s,f)\\,
(In fact, in the scalar case Cp = max{p — l,(p — I)" 1 }.) Indeed, when restricted to the probability space (/, |/|~ dx), equipped with the (7-fields induced by the sequence (£fc)fc>m, the sum in the definition of P(g, f) can be regarded (if we skip the terms with k < m which are constant on /) as the transform of the martingale difference sequence (Ak+ijj) by the adapted sequence (Ekf)Now, if g e BMOdiad(K) and 1 < p < oo, then
Let g — lj(g — c), where c is chosen so as to minimize the left-hand side of the above formula. Observe that 1 7 P(g, 1 7 / ) = 1 7 (P(<7, / ) + c') for some c'. Hence we can estimate
inf \\h(P(gJ)-c)\\p <
\\P(9,hf)\\,
If p 6 [l,oo) is fixed, then the BMOdiad(R)-norm of a measurable function h on R is dominated by sup /€ j( 2 ) |/|~ 'p infc ||lj (h — c)|| , hence the last estimate implies that
ll/lloo-
Figuel: Singular integral operators: a martingale approach
109
Furthermore, if / G L<x>(R) is a diadic atom supported by an interval / £ J ( l ) (i.e.,
1// = /, ll/lloo ^ W"1 ^d (/.I) = °): t h e n
we have
1/^/)
= f (*/) ^d a*80
(P(g, / ) , 1) = 0. Therefore, using Holder's inequality and the previous estimate, we obtain
\\P(9,f)\\x < \I\1/2\\hP(9,f)\\2 < 2|/|1/2inf Ui(P(s,f) ~ c)\\2 < \I\CU\L = C. This shows that the operator P(g, •), where P(g,-)(f) = P(g,f), is bounded from Loo(R) into J9MOdiad(R)) and from J5Tf fl
References [BS] C. Bennett and R. Sharpley, Weak type inequalities for Hp and BM07 Proc. Symp. in Pure Math. 35 (I) (1979), 201-229. [Bo] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences
are unconditional, Ark. Mat. 21 (1983), 163-168. Bo2] J. Bourgain, Vector-valued singular integrals and the H1 -BMO duality, Probability
Theory and Harmonic Analysis, J. A. Chao and W. A. Woyczynski, editors, Marcel Dekker (1986), New York 1-19. [Bu] D. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Prob. 9 (1981), 997-1011. Bu2] D. Burkholder, Martingales and Fourier analysis in Banach spaces, Springer-Verlag,
Lecture Notes in Mathematics 1206 (1986), 61-108. [C] Z. Ciesielski, Haar orthogonal functions in analysis and probability, Colloquia Math-
ematica Societatis Janos Bolyai, 49. Alfred Haar Memorial Conference, Budapest (1985), 25-56.
110
Figuel: Singular integral operators: a martingale approach
[CF] Z. Ciesielski and T. Figiel, Spline bases in classical function spaces on compact C°°
manifolds, II, Studia Math., 76 (1983), 95-136. [DJ] G. David and J.-L. Journe, A boundedness criterion for generalized Calder on-Zygmund
operators, Annals of Math., 120 (1984), 371-397. [F] T. Figiel, On equivalence of some bases to the Haar system in spaces of vector-valued
functions Bull. PoL.Acad. Sci. 36 (1988), 119-131. [M] Y. Meyer, Wavelets and operators, Proceedings of the Special Year in Modern Analysis at the University of Illinois, 1986-87, vol. 1, Edited by Earl R. Berkson, N. T. Peck Sz J. Uhl, Cambridge University Press, London Mathematical Society Lecture Note Series 137 (1989), 256-363.
Institute of Mathematics Polish Academy of Sciences ul. Abrahama 18 81-825 Sopot, Poland
Remarks about the interpolation of Radon-Nikodym operators
by N. Ghoussoub, B. Maurey, W. Schachermayer
0 Introduction : A bounded linear operator T between Banach spaces X and Y is said to be a Radon-Nikodym operator if it maps bounded X-valued martingales into converging Y-valued martingales. In [G-J], an example is given of an R.N.P operator that does not factor through an R.N.P space (i.e a space where the identity is an R.N.P operator). However, the question is still open in the case of a strong R.N.P operator i.e. when the closure of the image of the unit ball by the operator is an R.N.P set in the range space. This note consists of two parts. In the first, we shall prove that the interpolation method of [D-F-J-P] gives a positive answer to the above question if one considers what we call a controllable R.N.P operator ( See the definition below). Most of the commonly known strong R.N.P operators are of this type. However, in the second part of this note, we shall give examples which are not. Actually, we shall construct strong R.N.P operators for which the [D-F-J-P] factorization scheme gives Banach spaces that contain CQ. On the other hand, these operators are not counterexamples to the general problem because we do not know whether they factor through an R.N.P space by another scheme. It is well known that the above mentioned type of questions is equivalent to the following interpolation problem:
Suppose C is a closed circled convex bounded R.N.P subset of a Banach space
112
Ghoussoub et al: The interpolation of Radon-Nikodym operators
X. Can then one "embed" C in a Banach space with the R.N.P? In other words, if one considers the linear span Y of C equipped with the gauge ||.||c of C. Can one factor the injection j : Y —> X through a space with the R.N.P? We shall also deal with the analogous problem of interpolating P.C.P sets: i.e those whose all closed subsets contain points of weak to norm continuity. To justify the definitions below, we recall the following result proved in [G-M2]: A closed convex bounded subset C of a separable Banach space X has the P.C.P (resp the R.N.P) if and only if it is a strong w* — Gs (resp strong w* — H$) in some dual space, (i.e there exists a separable Banach space Z such that X can be identified with a closed subspace of Z*, in such a way that C \ C = UnKn with each Kn being it;*-compact (resp w*-compact and convex) in Z* and such that dist(A' n ,X)>0). For two sets G and H, let us denote by distm(G, H) = inf{dist(z, H); x 6 G} (resp, distj^(G, H) = sup{dist(x, H); x € G} the minimal (resp, the "maximal") distance between the sets G and H. We need the following definition: Definition (1): A closed convex bounded subset of a Banach space X is said to be a controllable P.C.P set (resp a controllable R.N.P set) if in the above representation as a w* — G$-set (resp w* — H^-set) the Kn's can be chosen in such a way that for some constant L > 0, we have for each n:
In other words, the oscillations of the distance to X on each Kn should be uniformly bounded. An operator T is then said to be a controllable P.C.P (resp a controllable R.N.P)
operator if the closure of the image of the unit ball by T is a controllable P.C.P
Ghoussoub et al: The interpolation of Radon-Nikodym operators
113
(resp R.N.P) set in the range. I. A positive result: In this section, we shall prove the following Theorem (2): IfT : X -> Y is a controllable RC.P (resp R.N.P) operator between two Banach spaces X and Y, then T factors through a Banach space E with the P.C.P (resp the R.N.P). The space E being the D-F-J-P interpolation space associated to the operator T. Proof: Let C = T(Ball(X) and let D = 0n[2nC + 2- n Ball(r)]. We need to show that D is also a P.C.P (resp R.N.P) subset of X. Before proving that claim, let us show how it implies the above theorem. First recall that if E is the interpolation space given in [D-F-J-P], and if T = SoR where R : X —> E and S : E —> Y, then we have that S is a Tauberian embedding; that is: S(Ball(£)) is a closed subset of D and S*(Ball(F*) is norm dense in E*. This necessarily implies that S is one to one. Note that if D is a P.C.P (resp R.N.P) set, then E has P.C.P (resp R.N.P) since S is then a nice Gs embedding [G-Ml], (resp a semi-embedding [B-R]). Actually, the weak topologies on Ball(jK) and its image by S are identical. Back to the claim, we shall prove that D is a w* — G$ (resp w* — H&) set in an appropriate dual space. For that, start by using the hypothesis on C to find a separable Banach space Z such that X is an isometric subspace of Z* in such a way that C \C = UnKn with each Kn being u>*-compact (resp iu*-compact and convex) and such that for some constant L > 0, we have distjt/(.K"n,.X") <
L.distm(Kn,X)
for each n. Let ~D* be the w*- closure of D in Z*. For each (n, k) £ N 2, define LU)k =
2k[Kn
114
Ghoussoub et al: The interpolation of Radon-Nikodym operators
Note that for each n, k the set Ln^ is tt;*-compact (resp tt;*-compact and convex). It remains to prove that
Note first that each L^ is disjoint from X so that Un Ujt A»,]fc C D \D. On the other hand, if y* G D \ D, we let 77 = dist(y*,X) > 0 and we choose &o such that 2~*° < 7//4L. We may write y* = y\ + yj with y\ G 2*°If and ||yj|| < 2"*°. As dist(yj, X) >-q - •%£> 77/2 > 0, there is an integer n0 such that 2~~*°yJ G Kno. This implies that distji/(A'no,X) > dist(2-fc°yJ,X) > 2~k°% and by our assumption, we get that distm(iifno,X) > ! Note now that
Ilv3ll< In other words y* = yi + v5 Thus proving the claim and the Theorem. Remark 3): One can also prove directly that the interpolation space E has the P.C.P (resp the R.N.P) by showing that it is a w* — G$ (resp w* — Hg) in the space F* obtained by the same interpolation applied this time to the io*-closure of C in Z*. II. The couterexample: In this section, we establish the following:
Theorem (4): There exists a closed absolutely convex R.N.P subset C of the unit ball ofco which is not a controllable R.N.P set. More precisely, the [D-F-J-P]-interpolation space associated to C contains an isomorphic copy ofco and hence fails to be an R.N.P space.
Ghoussoub et al: The interpolation of Radon-Nikodym operators
115
Proof: Let (an)n and (bn)n be two fixed positive sequences such that an | +00 and bn I 0. For each pair of sequences a = (an)n and p = (pn)n both going to zero, we shall construct a closed absolutely convex subset C(d, ft) of the unit ball of CQ such that: A) C(a, P) is a w* - H6 in €«, and therefore has the R.N.P. B) HO < pn < a»6 n andl > an > a" 1 , then the set D = nn(anC(a,P)
+
6nBall(co)) contains a subset that is affinely isometrically isomorphic to the unit ball of CQ and therefore the [D-F-J-P]- factorization space associated to C fails the R.N.P. First we introduce some notation. Let T = Uj^j-f—1, +1} be the dyadic tree and let 7 = T U T where T = {-1, +1} N . Let J = Uf=1 Jk with each Jk being the power set of{—1,+1}*. We shall denote by Pk an element of J that belongs to Jk. Let / be the countable index set / = I\ U I2 where /1 = N* = N \ {0} and / 2 = J x N* x N*. For i G /, we either write i = (n) to indicate that it belongs to I\ or we write i = (Pk, f, n) with Pk G J and (/, n) G N* to indicate that i G i^- The space co = co(/) will be defined over the index set I. Fix now sequences (a n ) n and (Pn)n both going to zero. For n G N* and (ei, ...,£m) G T, define the following elements of co(I):
an6j Pn !
0
if i G /1 and i = (j) for some 1 < j < m if i = (Pjb,/,n) G / 2 ,1 < / < m , l < A; < m and (£1, ...,£*) G . elsewhere.
For n = 0 and (ti, ...,em ) G T, define £j .0
if i G /1 and i = (j) for some 1 < j < m elsewhere.
It is clear that the above defined elements are finitely supported elements in CQ(I)
116
Ghoussoub et al: The interpolation of Radon-Nikodym operators
and their norms are HvA^i, •••»£m)|| = 1 while ||<£>n(£i, ...,e m )|| = an for every n G N* and any (ei,...,e m ) G T. Define now for each n G N, the sets An = {y>n(ei,...,£m);(£i,...,£m) 6 T} and Cn =closed convex circled hull of An. Let then A = U jj»i B U {0} and let C = C(a, /?) be the closed convex hull of U neN *C n . We will denote by A the w*closure in i^ of any subset A of CQ. We start by proving assertion (A). We divide the proof into four steps: Claim 1): For each n > 1, there exists a homeomorphic embedding (pn : T — : > £°°(I) such that n{T) = i n and v?n(T) = An = An n co(J) . Indeed, it is enough to take for (pn the continuous extension of
0
if i G /1 and i = (j) for some 1 < j < co if i = (ft, /,n) G / 2 is such that / G N* and ( d , ...,£*) G P;t elsewhere.
The proof of the following claim is obvious and is left to the reader. Claim 2): For n > 1, Pk G J, and x G An, the limit when / -> 00 of z((P*, /, n)) exists and is equal to either 0 or f3n. Moreover, the limit is /3n if and only if x is equal to
G F with ( £l ,..., e*) G P;fc.
For a signed measure /i supported on A, we shall denote by bar(^) the difference bar(/x+) - bar(/.<~) of the barycenters of the positive and negative parts of \i. Claim 3): a) We have: Cn = {bar(/i); fi
is a signed measure on (An, w*) with
\\/JL\\I
< 1}.
b) If // is a signed measure on (An,w*) with \\fi\\ < 1 and if x = bar(//), then the following assertions are equivalent: i) x G Cn = Cl4 nc 0 .
Ghoussoub et al: The interpolation of Radon-Nikodym operators
117
iii) For all P* G J, liminf/_^oox(P^,/,n) < 0 and liminf/^ 00 (-a;(P Jk ,/,n)) < 0. The proof of assertion a) of claim 3) as well as the implications ii) => i) => in) of b) are obvious. To prove that in) => ii), assume that fi and x — bar(^z) verify the condition iii). Suppose that ii) does not hold: that is |/i|(£ n (f \ T)) = \y\{(pn{Y)) > 0. To every P* G J, we associate the relatively clopen subset Vpk of (fn(T) = An \ An defined by
vPk = {¥>"((«;)£i); («;)£, e r
a^d(«,,...,«*) e A}).
Note that {Vpfc; P* G •/} forms a Boolean algebra generating the Borel cr-algebra of (An \ Anyw*). The assumption therefore implies the existence of some P* € J such that 77 = n{Vph) j£ 0. We can also suppose that 77 > 0 by passing to —// and — x if necessary. Find now/0 > k such that M({£ n(*);* € T, |t| > /o}) < r;/2. Since a; is the barycenter of ^, it follows that for every / > /o, x(Pf.,l, n) > 77/2 which is a contradiction.
Claim4): a) C = {ZZi A»*»;E«i IA«I < M « e Cn}b) An element x E C is in C if and only if for every n > 1 and P* G J, we have liminf/_>oo:r(Pjfc,/,n) < 0 and liminf/_,oo(—x(PkJ,n))
< 0. In particular, C is a
strong w* — H$ set in C and therefore it is an R.N.P set. Assertion a) follows from the fact that an — sup{||x||; x G Cn} tends to zero when n —• 00.
For b), let x G Cn and write x = J2Z=i Xn*n with Y%=\ IXn\ < 1 and xn ^ Cn> If each xn with An > 0 is in C n , then clearly x G C On the other hand, if there is no such that Ano > 0 and xnQ G Cno \ C no , then by claim 3), we may find P* G Jsuch that liminf/_^00xno(PA;,/,no) > 0 or liminf/_ 00 (-x no (Pjt,/,n 0 )) > 0. Noting that Ano£Wo(Pjt, I, no) = z(Pjfc, /, no) for every / G N* we obtain that
118
Ghoussoub et al: The interpolation of Radon-Nikodym operators
liminf/_+ooa;(Pjt, /, n 0 ) > 0 or liminf/^oo - x(Pk, /, n 0 ) > 0. This finishes the proof of b) and therefore part (A) of the Theorem. To prove part (B), assume that a " 1 < a n and 0 < f3n < anbn for every n. We claim that the set D = r\%Li(anC + 6 n .Ball(co(/))) contains Co which in turn can be identified with the unit ball of CQ. Indeed, we only need to show that
||a-V"(ei,...,e»)-V(ei,...,e»)l| = ^- < bn. This clearly proves assertion (B). To establish the theorem, it is enough to consider the sequences an = 2n, bn = 2~ n , an = 2" n , /?„ = 4- rt to conclude.
References
[B-R]
J. Bourgain, H.P. Rosenthal: Applications of the theory of semi-embeddings to Banach space theory. J. Funct. Anal. 52 (1983), p 149-187.
[B]
R.D. Bourgin: Geometric aspects of convex sets with the Radon- Nikodym property. Lecture notes in Mathematics, 993, Springer- Verlag(1983).
[D-F-J-P}V. J Davis, T. Figiel, W.B Johnson and A. Pelczynski: Factoring weakly compact operators. J. Funct. Analysis. 17.(1974), p.311-327. [G-Ml] N. Ghoussoub, B. Maurey: G$ -embeddings in Hilbert space. Journal of Functional Analysis.Vol.61, (1985) p.72-97 [G-M2] N. Ghoussoub, B. Maurey: H$ -embeddings in Hilbert space and Optimization on Gs -sets. Memoirs of the A.M.S. No.34 (1986).
Ghoussoub et al: The interpolation of Radon-Nikodym operators
[G-J]
119
N. Ghoussoub, W.B. Johnson: Counterexamples to several problems on the factorization of bounded linear operators. Proc. A.M.S 92, (1984), p 233238.
N. Ghoussoub
B. Maurey
W. Schachermayer
Department of Mathematics
U.F.R de Mathematiques Institut fur Mathematik
University of Bristish Columbia
Universite Paris VII
J. Kepler Universitat
Vancouver, B.C, V6T1Y4
75251 Paris Cedex 05
A-4040 Linz
Canada.
Prance.
Austria.
SYMMETRIC SEQUENCES IN FINITE-DIMENSIONAL NORMED SPACES W. T. Gowers Trinity College, Cambridge, CB2 1TQ, England §1. Introduction In 1982 and 1985 Amir and Milman published two important papers [2,3] showing that concentration of measure could be used to find large almost symmetric basic sequences in certain finite-dimensional normed spaces. In particular they exploited Azuma's inequality [4] and consequences of it due to Maurey [9] and Schechtman [11] in order to obtain the measure-concentration results they needed. In this survey we consider a number of properties of finite-dimensional normed spaces or finite basic sequences. Formally, we are interested in the following question. Given a space X of n-dimensions, let
122
Gowers: Symmetric sequences in finite-dimensional normed spaces
If under the same conditions one has only II
n
II
II
II
II
n
II
p r ^ a ; ^ ) < a ]T atxJ
II j
then the basis x\y...,
(1)
II
1
xn is asymmetric. A basis which is (1 + e)-symmetric for a
fairly small e is often said to be almost symmetric. If (1) holds whenever the permutation ?r is just the identity permutation, then the basis x\,...,
xn is said to be a-unconditional. A basis is also said to be almost
unconditional if it is (1 + e)-unconditional for a small value of e. The next definition is not standard, but it is a natural one in this context. Suppose a sequence of scalars « i , . . . , an is fixed. We shall say that a basis x\,...,
xn
is a-symmetric at a i , . . . , a n if (1) holds for the sequence a i , . . . , a n . If a is the vector ^2™ diXi we shall also say that x i , . . . , xn is a-symmetric at a. If the norm ||.|| being considered is not clear from the context, we shall sometimes say that x\,...,
xn is a-symmetric at a\,..., an or a under ||.||. We shall also speak loosely
of a basis being almost symmetric at a vector or sequence. Given a basis x\,...,
x n , a block basis is a sequence y i , . . . , ym where each
element yi is a vector of the form ^2j€A. ^jXj and the sets A\,...,
Am are disjoint.
It is more common to require also that if i\ < %2 and j \ £ AZl and J2 G A{2 then 7i < 32, but this definition is more suitable in a local context, and was the definition used by Amir and Milman. When the arguments of a function are specified, it should be understood that the function depends on these variables only. The results, presented here for spaces with real scalars, apply equally in the complex case. Finally, dimensions are often not written as integers, but all the results are true upon taking integer parts. §2. Positive Results The following two theorems may be found in [6,7]. Theorem 1. Let e > 0 , C > 1, l < p < o o and let # i , . . . ,xn
vectors in a normed space, satisfying the condition l/p
I. n
||
/
n
v
1/p
be a sequence of
Gowers: Symmetric sequences in finite-dimensional normed spaces
for every sequence a\,..., an of scalars. Then x\,...,
123
xn has a (1 + e)-symmetric
block basis u i , . . . , u m with ±1-coefficients and blocks of equal length, of cardinality m = a(e,p,C)n/ log n. Theorem 2. Let e > 0, 1 < p < 2 and let x\,...,
xn be a sequence of unit vectors
in a normed space, satisfying the condition 11
i=l
where the expectation is taken uniformly over all choices e i , . . . , en of signs. Then x\,...,
xn has a (1 + e)-symmetric block basis Ui,..., u m with ±l-coefRcients and
blocks of equal length, of cardinality m = a(e)n2lp~l / log n. These results improve on the bounds of a(e,p,C)n 1/3 and a(e)n (2 " p)2 / 2p3 obtained by Amir and Milman [2,3], and in fact, as we shall see in the next section, they are close to being best possible. Here we shall give a brief idea of how they are proved, and where the proof differs from that of Amir and Milman. Since the proofs of Theorems 1 and 2 are very similar, we shall concentrate on Theorem 1 in the case p = 1. Let n = mh and let (Rn, ||.||) be a normed space such that the standard basis e i , . . . , e n satisfies the conditions of Theorem 1 when p = 1. Define a random block basis Ui,...,u m as follows. First take a random partition of [n] into m sets A\,...,
Am of size h, where the set of all such partitions is given the uniform
distribution. Then pick a random sequence of signs (ei,..., en) € {—1, l}n> where again these are uniformly distributed. Let the block basis be u i , . . . , u m , where for each j the vector Uj is the restriction of (ei,..., en) to Aj. The main idea of Amir and Milman was to use concentration of measure to show that if h is large enough, then such a block basis will, with high probability, be almost symmetric. Our proof is also of this kind, but needs a smaller value of h. Given any sequence a = ( a i , . . . , a m ) G Rm, let a be the random vector ZIjli ajuj-
By a standard application of a general result of Schechtman [11], or
directly from Azuma's inequality, one can show that P [| ||a|| - E ||a|| | ^ S ||a||] < exp(-* 2 ||a||2 /sC2 ||a||2) .
(2)
124
Gowers: Symmetric sequences in finite-dimensional normed spaces
Now, given any vector generated by the random block basis, the ratio of its ^i-norm to its ^-norm is at least \fh.
Using this estimate in (2), fact that a
norm is controlled by its behaviour on a sufficiently fine net of its unit sphere, and the fact that, given the unit sphere of an m-dimensional normed space, it has a 6-net of cardinality at most (1 + 2/£) m , one can show that u i , . . . , u m has a high probability of being (1 -f- e)-symmetric provided h ^ /?(e, C)™1/2, and thus whenever m ^ a(e, C)nlf2. In order to improve this estimate to a(e, C)n/logn, the main idea is to consider various classes of vectors separately. Note that the estimate ||a||j / ||a||2 > y/h is very weak if for example a = ]CfLi uj- Eliminating this and similar weaknesses leads to our improved result. The proof divides naturally into two parts, which we shall not prove. Lemma 3. Let 8 > 0, let (R m ,||.||) be a normed space and set N = m 6 " 1 * 0 ^ 3 *" 1 ). There exist N vectors a i , . . . , aw such that if \\.\\ is (1 + 8)-symmetric at a,- for every i, then the standard basis of Rm is (1 + S)(l — 68)*1-symmetric. Lemma 4. Let e and C be as in the statement of Theorem I, let 6 = e/11 and let h = ^(e,C)logn. Let a = ( a i , . . . , a m ) € R m be a given vector with Ha^ = 1, and for (r),a) <E { - l , l } m x Sm let aT/)<7 denote the vector (771 a f f ( 1 ) ,...
,iqma(r^rn)).
Then
The theorem follows easily from these two lemmas. Indeed, suppose they are both true and pick a sequence a i , . . . , a w as guaranteed to exist by Lemma 3, taking 6 = e/11. By Lemma 4, the probability that the random block basis fails to be (1 + <$)-symmetric at any given vector a is less than N~1. It follows that it has a positive probability of being (1 -f- £)-symmetric at each of a i , . . . , a^. But then, by Lemma 3, there is a positive probability that it is (1 + e)-symmetric, since
The important part of the proof is of course Lemma 4. Since it concerns the deviation of several vectors at once, one might expect the proof to use a result
Gowers: Symmetric sequences in finite-dimensional normed spaces
125
about sub-Gaussian processes. However, it turns out that this is not necessary. For example, if a is the characteristic function of a subset of [m], then Lemma 4 is a trivial consequence of (2): the main difficulty in proving Theorem 1 is in fact the technical problem of coping with vectors with coordinates of widely differing sizes. To prove Lemma 4 for such vectors, one splits them into parts with approximately equal coordinates and applies the triangle inequality rather crudely. Details as well as a proof of Lemma 3 may be found in [7]. It is natural to ask whether a basis equivalent to the unit vector basis of ^ must have a large almost symmetric block basis. The following result is the best that is known in the positive direction. Theorem 5. Let x\,... ,xn be a sequence C-equivalent to the unit vector basis of ££>. Then it has a block basis of cardinality at least k = na, where a = log(l -|- e)/21og C, which is (1 + e)-equivalent to the unit vector basis of ^Jo. This can be proved by a simple technique, essentially due to James. We shall see in the next section that in a sense one cannot do much better than this. Another very natural question is the following. Suppose X is an arbitrary n-dimensional normed space. How large a (1 -f- e)-symmetric basic sequence must it contain? The following result is due to Alon and Milman [1]. Theorem 6. Let 0 < e < 1, let X be an n-dimensional normed space and let m = exp(cyje~l logn), where c is an absolute constant. Then X contains an m-dimensional subspace Y such that either d{Y,V?) < 1 + e or d{Y,^Q) < 1 + e. As remarked by Alon and Milman, this result, as it stands, is best possible (simply consider ££ for appropriate p = p(n)). However, if one is looking only for a subspace with a (1 -f e)-symmetric basis, it is not known whether the result can be substantially improved. An upper bound will be given in the next section. The corresponding question about unconditional sequences is also interesting, and Theorem 6 seems to be all that is known in either direction. Finally, we mention a result of Amir and Milman [2] concerning unconditional block bases of arbitrary bases.
126
Gowers: Symmetric sequences in finite-dimensional normed spaces
Theorem 7. Let e > 0 and let xi,...,xn vectors in a normed space. Then x\,...,
be a linearly independent sequence of xn has a (14- e)-unconditional block basis
1 2
of cardinality at least a(e)(logn) / .
§3. Negative Results In this section we shall give an idea of how one constructs sequences and spaces to give upper bounds corresponding to the results in the last section. To begin with Theorem 1, we have the following counterpart. Theorem 8. Let 1 ^ p < oo and e > 0. For every n 6 N there exists a sequence x\,...
,xn which is 2-equivalent to the unit vector basis of£p and which has no
(1 + e)-symmetric block basis of cardinality exceeding mo, where ( a(e,, p)n p) n log log nnj log n mo
1 < p < oo
\p(e)n/log\ogn
and a(€,p) —>0ase—* 0 or p —• co, and /9(e) —> 0 as e —• 0. The method of proof of Theorem 8 is in a sense the reverse of the method of proof of Theorem 1. In Lemma 3 we found as few vectors as possible with the property that their norms, and the norms or their rearrangements, control the norm on the whole space. The counterpart to Lemma 3 is the next lemma. We shall not give the exact definition of a standardized block basis, since it is only needed for technical reasons. The two most important points are that any vector in a standardized block basis is supported on at most 2n/mo points, and that any block basis contains a subset which is a multiple of a standardized block basis. Lemma 9. Let 0 < e < 1/4, 1 < p < oo, m = n 3 / 4 and M = n«(p)/i°giogn. exists a sequence of subsets A\,...,
Tnere
AM of the unit sphere of£^ with the following
properties. (i) For any 1 ^ j ^ M and any standardized block basis u i , . . . , u m there exists a sequence of scalars a i , . . . ,a m such that Y^T a i u » € A r (ii) Let 1 < j < My let u i , . . . , u m be a standardized block basis and let a i , . . . ,a m be a sequence for which Y^T aiui € Aj. Then, for any permutation
Gowers: Symmetric sequences in finite-dimensional normed spaces
7T € Sm and any sequence e\,...,
127
e m of signs, m
1
(iiij Let 1 ^ j < k < M, iet / be the £™-support functional of a vector in Aj and let a € A*. Then |/(a)| < (1 - 4e) ||a|| p . Lemma 9 is useful because it enables one to construct a norm by constructing its restrictions to the sets A\,...,
AM separately. Given any 1 ^ j ^ M it turns
out to be easy to construct a norm randomly in such a way that its restriction to Aj has two useful properties. First, it is (1 -f- 2e)-equivalent to the usual tpnorm, and second, any given standardized block basis Ui,..., u m has a very small probability of being (1 + 2€)-symmetric at the vector in Aj which, by Lemma 9 (i), it must generate. (Note that, by Lemma 9 (ii) all the rearrangements of this vector are also in Aj). Let us write Sj for the event that the given block basis is symmetric at the vector it generates in Aj. By far the most important part of Lemma 9 is the third part. This enables us to put together the various restrictions to the sets Aj to give a random norm with the property that the events Sj not only have small probability but are also independent. That is, the restriction of the norm to Aj does not interfere with its restriction to Ak for k ^ j . Hence, the probability of a given block basis being (1 + 2e)-symmetric is at most II^ :1 P(5j). It remains to pick L standardized block bases which form a sort of net, for some L that is not too large. They have the property that if they all fail to be (1 + 2e)-symmetric, then every standardized block basis (and hence every block basis) fails to be (1 -f e)-symmetric. This can be done with L ^ (JIjfL1P(Sj))'~1. Hence, with a non-zero probability, the random norm has the property that no block basis of cardinality mo is (1 + e)-symmetric. When p = 1, the unit sphere of £JJ is not sufficiently convex to accommodate a large number of classes of vectors on which a norm can be defined independently. A rather delicate construction can be used in a space which is C-equivalent to ££, but even this yields many fewer classes than are obtained when p > 1. That is
128
Gowers: Symmetric sequences in finite-dimensional normed spaces
why the upper bound is larger. It seems at least possible that this bound is the correct one, rather than nj log n, but it would probably be hard to prove this. There is an easy upper bound for Theorem 2. Set m = 2n2/p~1, h = n/m, let e i , . . . , e m be the standard basis of £™ and, for 1 < i < n, let #,- = ej-j/^i. It is not hard to verify that the conditions of Theorem 2 are satisfied, and since the vectors x\,...
,x n are in a space of dimension m, there obviously cannot be an
almost symmetric block basis of cardinality greater than m. The following result is more interesting, since the sequence constructed satisfies a lower p-estimate. Theorem 10. There exists an absolute constant C such that, for any 1 < p < 3/2 and any n £ N, there is a norm \\.\\ on Rn satisfying the following three conditions: (i) the standard basis is normalized; (ii) for any a e Rn, ||a|| > ||a|| p ; (Hi) if k ^ Cn2/*""1 (log n) 4 / 3 then no block basis uu . . . , uk of the standard basis is 2-symmetric.
The construction is a fairly simple random one. For suitable N (a power of n will do) one picks N functionals / i , •.., //v randomly from the 2 n functionals with ±l-coordinates, distributed uniformly. Then the random norm on Rn is given by llxll = max |/,(x)| V Ibll . The proof that this construction works is quite long, but it is considerably simpler than that of Theorem 8. It fails when p > 3/2, because one can, with high probability, find a block basis which is close to the standard basis of ^ , where m is approximately n 1 / 3 . We shall discuss sequences which satisfy a lower 2-estimate later in the section. The next example is an explicit construction which can be adapted to give upper bounds for several of the problems mentioned so far. The following result is the most important one connected with the construction. It is proved in detail in [8], as are Theorems 12-14.
Gowers: Symmetric sequences in finite-dimensional normed spaces
129
Theorem 11. Let M > 1. Then there exists an absolute constant c such that if n ^ f(M) is sufficiently large, then there exists an n-dimensional normed space X containing no M-symmetric basic sequence of cardinality exceeding n c / l o 8 l o 8 n . Theorem 11 gives a negative answer to a question of Milman. He asked (private communication) whether every n-dimensional space contains a 2-symmetric sequence of cardinality proportional to y/n. It is also relevant to the dependence on e in Dvoretzky's theorem. Bourgain and Lindenstrauss [5] have shown that there exists a function f(e,k) which is polynomial in e"1 for fixed &, such that any space X of more than /(e, k) dimensions with a 1-symmetric basis has a kdimensional subspace Y such that d(Y,$2) ^ 1 + e. Lindenstrauss has informed me that this is also the case if X has a 2-symmetric basis. Hence, if there were some fixed 7 > 0 and c > 0 such that any n-dimensional normed space contained a 2-symmetric basic sequence of cardinality at least en1', then the dependence on e in Dvoretzky's theorem would be polynomial for arbitrary n-dimensional spaces. Although Theorem 11 shows that this premise is false, our construction does not give an example of non-polynomial dependence for Dvoretzky's theorem. The example is constructed as follows. Given any space X = (R m, ||.||) such that the standard basis is 1-unconditional, let £p(X) denote the p-direct sum of k copies of X. Then, setting p = 12, k = log log n/2 log 12 and m = n1 /*, the space X is
This result cannot be improved by choosing a different iterated direct sum of ^,-spaces. It is not hard to deduce from Theorem 2 that any such space contains a (1 + e)-symmetric sequence of cardinality 0 1 (e)n c /* 2 ^ +loglogn , for an absolute constant c. Using the same construction, or simple variants of it, the following results can also be obtained. Theorem 12. Let e > 0, C > 1, let n £ N and set a = log(l + e)/logC and k = Iog 12 (l/8a). Then there exists a basis C-equivalent to the unit vector basis with no (1 + e)-symmetric block basis of cardinality n8/*(log2 n)*.
130
Gowers: Symmetric sequences in finite-dimensional normed spaces
Theorem 13. Let M ^ 1 and let n be a power of 2. Then there there exists a normalized basis x\,...,
xn satisfying a lower 2-estimate with no M-symmetric
block basis of cardinality n40/loglogn Theorem 14. There exist absolute constants c,c' such that, for every q > ec and M ^ 1, and for n sufficiently large, there exists an n-dimensional space with q-cotype constant at most c'', with no subspace of dimension n c / l o g g with an Msymmetric basis. Theorem 11 can almost certainly be improved. Any improved construction could also be used to improve Theorem 13, and probably adapted to improve Theorems 12 and 14 as well. It seems likely that the result of Alon and Milman (Theorem 6) is the best possible positive result concerning the existence of symmetric sequences in arbitrary spaces. A proof of this would be extremely interesting. §4. Open Problems We shall finish the paper by choosing some of the more interesting problems arising from the previous sections. Problem 1. Let C > 1, let e > 0 and let xi,... ,xn be a sequence of vectors equivalent to the unit vector basis oft™. Does x\,...,
xn necessarily have a (1 -h e)-
symmetric block basis of cardinality a(e, C)n/ log log n ? Problem 2. Let X be an n-dimensional normed space. How large a 2-symmetric basic sequence must it contain? In particular, does it necessarily contain such a sequence of cardinality substantially larger than exp(y/\ogn)? The third problem is the same as Problem 2 with "2-unconditional" replacing "2-symmetric". Problem 3. Let X be an n-dimensional normed space.
How large a 2-
unconditional basic sequence must it contain? In particular, does it necessarily contain such a sequence of cardinality substantially larger than exp(\/logn)?
Gowers: Symmetric sequences infinite-dimensionalnormed spaces
Problem 4. Let xi,...,
131
xn be any sequence of linearly independent vectors. Then
how large a 2-unconditional block basis does it have? In particular, does it have a 2-unconditional block basis of cardinality significantly greater than y/logn? Problem 5. Let xi,...,xn
be any sequence of linearly independent vectors. Then
how large a 2-symmetric block basis does it have? There does not seem to be strong evidence in either direction for Problems 1 and 3. It does seem quite likely that Problem 2 has a negative answer and Problem 4 a positive one. If Problems 2 and 3 or Problems 4 and 5 could be shown to have different answers, it would be the first non-trivial example of a property of a space or basis which makes it significantly easier to find an unconditional sequence than to find a symmetric one. In connection with Problem 3, it is probably possible to adapt Theorem 10 to give an upper bound of na for some fixed a < 1, and it is definitely possible to do this for Problem 4. In fact, for each n £ N, there exists a basis of cardinality n with no 2-unconditional block basis of cardinality greater than C?21/3(log?i)4/3, where C is the absolute constant in Theorem 10. This upper bound is of course much larger than the lower bound given by Theorem 7. The best known lower bound for Problem 5 seems to be given by Krivine's theorem, so it must be possible to improve it. References [1] N. Alon and V. D. Milman, Embedding of i^ in finite dimensional Banach spaces, Israel J. Math. 45, 265-280. [2] D. Amir and V. D. Milman, Unconditional and symmetric sets in n-dimensional normed spaces, Israel J. Math. 37 (1980), 3-20. [3] D. Amir and V. D. Milman, A quantitative finite dimensional Krivine theorem, Israel J. Math. 50 (1985), 1-12. [4] K. Azuma, Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), 357-367. [5] J. Bourgain and J. Lindenstrauss, Almost Euclidean sections in spaces with a symmetric basis, GAFA 87-88, Springer Lecture Notes 1376 (1989), 278-288.
132
Gowcrs: Symmetric sequences in finite-dimensional normed spaces
[6] W. T. Gowers, Symmetric block bases in finite-dimensional normed spaces, Israel J. Math., to appear. [7] W. T. Gowers, Symmetric block bases of sequences with large average growth, Israel J. Math., to appear. [8] W. T. Gowers, Symmetric structures in Banach spaces, Ph. D. thesis, University of Cambridge. [9] B. Maurey, Construction de suites symetriques, C. R. A. S., Paris, 288 (1979), 679-681. [10] V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Springer Lecture Notes 1200 (1986), viii 4- 156 pp. [11] G. Schechtman, Levy type inequality for a class of metric spaces, Martingale Theory in Harmonic Analysis and Banach Spaces, Springer-Verlag 1981, 211215.
Some topologies on the space of analytic self-maps of the unit disk Herbert Hunziker, Hans Jarchow*and Vania Mascioni Department of Mathematics, University of Zurich Ramistrasse 74 CH-8001 Zurich (Switzerland)
Abstract We study some topologies on the space $ of analytic self-maps of the unit disk induced by composition operators on H2. Among these is the so-called Hilbert-Schmidt topology, for which we find that the connected components are always arc connected, that they coincide with the sets {tp : Cxf, — dp is Hilbert-Schmidt}, and that they are convex subsets of the unit ball of H°°. These properties are also shared by the other topologies under consideration, and all of them turn out to be intimately related to the classes of composition operators which are order bounded as maps H2 —* Lp .
1
Introduction
Let D be the open unit disk in the complex plane. Consider the class $ of all analytic functions which map D into D, that is, the unit ball of the Hardy space H°° without the constant functions generated by elements of dD . Each ip G $ is well-known to give rise to a composition operator C v on the Hardy space H2 (see §2 for all definitions), and the usual operator norm || • || on B(H2) induces a metric on $ defined by
Let us denote the space thus obtained by ($, | * Supported by the Swiss National Science Foundation
134
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
In [6,8], MacCluer, Shapiro and Sundberg have studied the connectedness properties of (<3>, || • ||). Their research was mainly motivated by a result of Berkson [1], according to which
(note that the distance between two points may be infinite). Let us agree to call this new topological space ($, 2). We will see that the connected component of (p in ($,2) is exactly the set {tp : C^ — C
2
Notation
Let H2 be the Hilbert space of all holomorphic functions in the open disk D having square summable Fourier coefficients. If / € H2 then / has radial limits /(C) = linir^! /(r(") for almost all £ £ 3D, and the scalar product of / , g £ H2 is given by a being the normalized Lebesgue measure on dD.
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 135
More generally, if 0 < p < oo then Hp is the (p—)Banach space of all analytic functions f on D such that the integrals /02* \f(reie)\p dO, 0 < r < 1, are bounded. H°° is the Banach space of all uniformly bounded holomorphic functions on £), the norm being given by ||/||//°° = ess sup<edD |/(C)I • In the sequel, we shall freely identify functions in Hp with their radial limits. Writing / € Lp, we will always mean that the function f on D has radial limits belonging to Lf^dD, a). To simplify the notation, we will write $ for the unit ball of H°° without the constant functions generated by points in 3D. It is a well-known consequence of Littlewood's Subordination Principle [5] that each (p in $ gives rise to a continuous composition operator C^ : Hp —• Hp, / »-> / o ip (0 < p < oo). It is clear that
where (t/)n)n is the canonical orthonormal basis of H2 given by ^n(C) = CnBy Shapiro and Taylor [9], a composition operator C v on H2 is Hilbert-Schmidt if and only if (1 - M)" 1 € L1. If C G D is fixed, the composition operator corresponding to the function in $ taking the constant value £ may be identified with the evaluation functional 6$ at C: so £ c (/) = /(C). On $ we can define the [0, oo]-valued "metric" by exp {I fdD log ||^ ( c ) - 6H0\\ da(Q}
,
p =0
, 0 < p < oo ,
p = oo
where all norms are taken in (H2)*. We will denote by ($,p) the space $ endowed with the topology generated by the open sets
It is easy to verify that dp/(l + dp) is a metric on (^,p) if p > 1. Also, it follows easily from the definition that the identity ($,p) —> ($, q (if p < q this does not hold, as we shall see after Proposition 3.5). Further on, we will show that ($,2) inherits its topology from the HilbertSchmidt norm for operators in H2.
136
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
Given
L
JdD
We shall use standard notation and terminology from Banach space theory.
3
The spaces (<&,p)
Let us begin with an easy lemma. Lemma 3.1 Let
(ii)
log |y>-|, l o g | l - ^ |
eL1.
The following are equivalent: (a)
P r o o f (i). Since
holds, and so we get log |1 — (pip\ £ L1. (ii). By (a), log(l - |v?|),log(l - |^|) G Ll, and so (b) follows easily. The implication (b) => (a) is clear and (b) <=> (c) follows from this, using (i) and the inequalities
i-MW - M 2 )(l -
^
i
^ 1 / 2
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 137
Theorem 3.2 Let
belongs to Lp/2, case
resp.
if and only if log !(<£>, xp) G Ll when p = 0, and in such a ex
P {I IdD lo S ^(V3, tyd*} , ( / a D ^ ^
Proof
P=0
/
ess sup a D J(y>,
! Jn particular,
1
if ip ~p x(> and
then neither ip nor xp is in $ e x t .
First note that, if 77, ti G D and / G H2, then
where ^(C) = 1/(1 — ^ 0
IS
^
e
reproducing kernel for H2 . We may thus write
1
1
:
+ «—r
i-M
1 It is easy to see that, if ip ~p xp and
Remarks (1) If C
138
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
which is readily seen to be equal to ^ ( ^ j ^0- This explains why we propose to call the topology on $ induced by d2 the "Hilbert-Schmidt topology". (2) If (p, xj) € $ we observe that
i - M2
i -
we may deduce from the last inequality in the proof of Lemma 3.1 that, in general, 2 —
I-(pip
Proposition 3.3 ($,p) is complete for all 0 < p < oo. Proof Let (ipn) be a Cauchy sequence in ($,p). Thus, given e > 0, we can find an Ne such that if m,n > Ne then
Let first p > 0. Since /(<^n,(^m)1/2 > \ipn — <^m|, we see that (y?n) is a Cauchy sequence in Hp and so has, in particular, a pointwise limit Ne , which shows that (y>n) converges to (p in (^,p). If p = 0 , subharmonicity of log \
so that ((fn) converges pointwise. Now we proceed as in the case p > 0 . Theorem 3.2 enables us to characterize the isolated points in ($,p):
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 139
Theorem 3.4 Let
(in)
(p is an isolated point in ( $ , p ) .
Proof If cp 6 ^ext then dp((p, xp) cannot be finite for any xp =^ (p in $ , by Theorem 3.2. By definition this implies Kp((p) = {|)3/2 o n 3D - K w e define
for all —1 < 5,t < 1. This shows, by Theorem 3.2 and dominated convergence, that < H ^ t actually defines a continuous arc in (4>,p) passing through
By [8], not all extreme points of $ generate isolated composition operators in ($, || • ||) • In fact, there is even a (p 6 $ ext such that C^ is compact: however, all compact composition operators belong to the same (arc) connected component in
(*.IHDIn the next proposition we collect some additional information. We denote by 0 the zero-function in $. Proposition 3.5 Let 0 < p < oo. (i) (p e A'p(0) if and only if
(
,
(1 - M)-i € L*'2
,
\W\\H<*> < 1
,
P= 0
0
fnj Zei ip ~p ifi. Then the following are equivalent:
(a) v,4,eKp(p) (b) (iii)
oo
140
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
Proof (i) is a direct consequence of Theorem 3.2. (ii). If p = 0 then the assertion is trivial. Let next 0 < p < oo. By Theorem 3.2, ip ~ p i/> if and only if I(y>, tf) € L p / 2 . By (i), ip € Kp(0) implies (1 - M ) " 1 G Lp / 2 , and thus the smaller function (1 — ly^l)" 1 is m Lp^2 <> too. Conversely, if (1 - l ^ l ) " 1 e L p / 2 , then £[1/(1 - V)] € £ p / 2 , since
Since
I(y>,^) € £ p / 2 implies that
a.e. on 3D. Writing \tp - ?/>|2 = |1 - (pip\2 - (1 - |y>|2)(l - M 2 ) , it is immediate to deduce that K +
a.e. on dD. Consequently, we must have ||
i - M 2h/f 2
(l - M )(i - h/f)
=
i
,
=
i-M
2
2 2
2
we easily get
i ih/f
r
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 141 whence (1 — j£^L
)
£ Lpl2, which was what we wanted. D
Remark The description of Kp(0) given in Proposition 3.5(i) allows to conclude that the topologies ($,p) are in fact all different. To see this, it suffices to find a function (p in
f]Kq(0)\Kp(0)
,
which can be done as follows: take a real function / > 2 on 3D such that
fe f| Since log(l — f~l) G Ll, Szego's Theorem [3, p.53] provides us with a function
vr.>v*A
*/
^^
U(15)
t
{
1
t
]
U
2(1 - cosi?) [ ( 3 - t ) 2 + 2 5 ( 1 - 5 ) ^ ( 1 - < ) ( ! - c o s t9)] '
and this belongs to Lp^2 only if p < 1/2. Consequently, if p > 1/2 then each (pt must lie in a different component of ($,p). However, if 0 < t < 1, these operators cannot be isolated in ($,p) for any p: in fact, they fail to be extreme points of <£ (see Theorem 3.4).
142
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
The following proposition exhibits another elementary but somewhat surprising property of the sets Kp(0): the straight line connecting any (pi £ $ to any (p0 6 Kp(0) consists almost entirely of elements of KP(0). Proposition 3.6 Let 0 < p < oo, y>0,¥>i £ $ • If ¥>o € # P (0), then ipt e KP(Q) for
allO
Proof If p = 0, oo, the proposition is trivial. If 0 < p < oo, just note that, for 0 < t < 1, < < and that the last function is in Xp/2, by ipo € KP(0) and Proposition 3.5(i).
4
Connected components in ($,p)
Let us first prove an elementary inequality for points in D. Lemma 4.1 Let ao,a,i G D and at = (1— t)aQ+tai for 0 < t < 1. Then, regardless ofQ<s
- K1 2 M 2 (1 - |a.P)(l < 2
- aQax
Proof Without loss of generality, we may assume 5 = 0. Note first that a
o ~ at I2 - doat I
t \&o ~ ^i|
- |a o | 2 ) (1)
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 143
On the other hand, since the function t <-+ (1 — la^2 )"1 is convex,
(1
||»)(l
| P )
+
1 ||» 1 - |«o|*
1
k| 2
1 -
The assertion follows by combination of (1) and (2).
Corollary 4.2 Let 0 < p < oo. If <po ~p
,
0 < t < 1
provides a continuous arc in ($,p) connecting (po and
Define (pt = (1 — t)<po -f t
holds pointwise for all 0 < «s,t < 1. The corollary follows from Theorem 3.2 and Lebesgue's dominated convergence theorem. D
Summarizing all the preceding results, wefinallyarrive at a complete description of the connected components of ($,p). Theorem 4.3 Let 0 < p < oo. For each ip E $ , the set is the connected component of ($,p) containing (p. Kp(ip) w always arc connected in ($,p) arc-d convex in H°°. Kp(
144
5
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
(r,p)-order boundedness and (<£,£>)
Let 0 < r,p < oo. We say that an operator T: Hr —» Hr is (r,p)-order bounded if there exists a function h G Lp(dD) such that |(T/)(C)| < h(0 a.e. on 3D for all / G BHr. Thus (r, p)-order boundedness of T means that T can be regarded as a bounded operator T : HT -* Hp, and that JPT : Hr -> Lp(dD) is order bounded in the traditional sense, J p being the natural embedding Hp^Lp(dD), or majorizing, in the terminology of [7] (p > 1). The following theorem characterizes (2,p)-order boundedness of composition operators in terms of the topological spaces ($,p): Theorem 5.1 Let 0 < p < oo and ip,xp G $ . T/*,en C v — C$ is (2,p)-order bounded if and only if ip ~ p tf>.
Proof Let / G Bfp and ( G D. Then we always have
(/)l (1) Thus, if y? ~p %j), then /(v?,^) 1/2 G Xp by Theorem 3.2, and so /(v?,^) 1 / 2 acts as an order bound for the functions in (Cv — C,/,)(J?//2), i.e. we get the (2,p)-order boundedness of C^ — C^ . Conversely, if C^ — C^ is (2, p)-order bounded, and if h G Lp is such that |(CV - Q)(/)(C)| < h(0 a.e. on dD for all / G JB//2 , then we get /(~p. D
Combining this with Proposition 3.5, we get immediately Corollary 5.2 Let 0 < r,p < oo and
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 145
(Hi)
Proof (i) <$ (ii) : The case r = 2 follows immediately from the above theorem and from Proposition 3.5. If r ^ 2, just use the fact that
for all 77 £ JD , and argue as above. (ii) <& (iii) : It follows from
and from that 1/(1 - \(p\) belongs to Lp/r if and only if
which means (||y?n||//i) G £r/P,i (iii) <& (iv) is trivial. Remarks (1) In particular, we see that C
146
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
(3) Let y?, V> £ $ , 1 < r < oo and 1 < p < oo be such that Cv + C+ is (r,p)order bounded. Through harmonic functions, it is not difficult to see that Cv + C^ can even be regarded as an order bounded operator U —> Lp. Since composition operators defined on Lr-spaces are trivially positive, C^ and C^ must both be (r,p)-order bounded. Taking r = p = 2 and observing that C^ -f C^ is (2,2)-order bounded if and only if C v -|- C^ is Hilbert-Schmidt on if2, we obtain that C^ -f- C^ is Hilbert-Schmidt on H2 if and only if both C^ and C^ are Hilbert-Schmidt. Generalizations of this will be discussed elsewhere. Our final theorem exhibits a somewhat unexpected "ideal" property of the components of ($,p): Theorem 5.3 Let 0 < p < oo and let x G $ &e arbitrary. Then
In particular, if ip (z Kp(0), which x € $ .
then <pox and XO(P belong to Kp(0),
for no matter
Proof The case p = 0 follows easily from Proposition 3.5(i). Let 0 < p < oo,
p
and that /(<,£>, ip) ^ (z L . As a consequence, we get that a.e. on Now, via appropriate spaces of harmonic functions, Cx induces a continuous operator on Lp , and thus /(y>, VO^^X ^ Lp appears as an order bound for the functions in (Cvox - CVK>x)(jBi/2). Hence, yox ~p ox.
a Remarks function x
(1) To conclude, let us briefly look at what happens if our composition Mobius transform. Given r\ £ Z), write
1S a
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk 147
and consider the map *n : ($,!>) -> ($,P) ,
1
. „
. . 5-l»?l 2
1
and so y> ~ p r^oy? holds if and only if
we get that the map ^ is a homeomorphism of (^,p) onto itself. Moreover, in induces a permutation of the components of ( $ , p) , with "fixed component" A"p(0). (2) Concerning composition with V'n(C) = Cn ( n ^ 1)? w e c a n prove that (for 0 < p < oo)
(i)
jKp(O)
(ii) ^ o ^ n - p ^ o ^ n ^ ^ ~ p ^ (iii)
(^ G J^p(O) <^> n o ^ G A' p (0) .
Details will appear elsewhere.
References [1] E. BERKSON, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981) 230-232. [2] O. BLASCO, Positive p-summing operators on Lp spaces, Proc. Amer. Math. Soc. 100 (1987) 275-280. [3] K. HOFFMAN, Banach spaces of analytic functions, Prentice-Hall, 1962. [4] H. HUNZIKER, Kompositionsoperatoren auf Hardyraumen, Thesis, Univ. of Zurich, 1989.
148
Hunziker et al: Topologies on the space of analytic self-maps of the unit disk
[5] J. E. LlTTLEWOOD, On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925) 481-519. [6] B. D. MACCLUER, Components in the space of composition operators, Preprint. [7] H. H. SCHAEFER, Banach lattices and positive operators, Springer Verlag,
Berlin 1974. [8] J. H. SHAPIRO AND C. SUNDBERG, Isolation amongst the composition operators, Preprint. [9] J. H. SHAPIRO AND P. D. TAYLOR, Compact, nuclear, and Hilbert-Schmidt composition operators on Hp, Indiana Univ. Math. J. 23 (1973) 471-496. [10] H. J. SCHWARTZ, Composition operators on Hp, Thesis, Univ. of Toledo, 1969.
Minimal and strongly minimal Orlicz sequence spaces N . J . KALTON* DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI COLUMBIA, MO. 65211
(*): This research was supported by NSF-grant DMS 8901636. 1. Introduction. The structure theory of Orlicz sequence spaces was initiated in work of Lindberg [6] and Lindenstrauss and Tzafriri ([7], [8] and [9]; see also [10]) in the early seventies. Recently this study has been continued by Hernandez and Rodriguez-Salinas ([2], [3]). In their work, Lindenstrauss and Tzafriri introduced the class of minimal Orlicz sequence spaces and conjectured that these spaces are prime. The only separable prime spaces known are the lp and c 0 , but Lindenstrauss and Tzafriri gave other examples of minimal Orlicz spaces. The purpose of this note is to show that this conjecture is false in general, but that a smaller non-trivial class of strongly minimal spaces is introduced which still has the potential to contain new prime spaces. Our results are achieved by introducing separate necessary and sufficient conditions for a reflexive Orlicz sequence space to to be complemented in another such space lp. We now proceed to a more detailed discussion of the basic definitions and our results. We refer to [10] for the basic facts about Orlicz sequence spaces, but review here some key definitions and ideas. It will be convenient to allow Orlicz functions to be possibly non-convex, even though we do not wish to discuss non-locally convex examples. Thus we will for the purposes of this note consider an Orlicz function to be a continuous function F : [0,oo) —• [0,oo) satisfying F(0) — 0, such that F(x) > 0 if x > 0 and satisfying, for a suitable constant C, F(tx) < CtF(x) whenever 0 < t < 1 and 0 < x < oo. We say that F satisfies the A 2 -condition if F(2x) < KF(x) for a suitable constant K and all x. Two Orlicz functions F and G are equivalent if \ogF(x)/G(x) is bounded on (0, oo) and equivalent near zero if log F(x)/G(x) is bounded on (0,1). Any Orlicz function satisfying our definition and the A2-condition is then equivalent to a convex Orlicz function. If F is an Orlicz function satisfying the A 2—condition then we define the Orlicz function space Lp — Ljr(0,oo) to be the space of all measurable real functions / on (0,oo) satisfying '°°F(|/(t)|)
150
Kalton: Minimal and strongly minimal Orlicz sequence spaces
only F and G are equivalent near zero. Lp(0,00) is reflexive if and only if there exists lF is a > 0 and a constant C so that if 0 < t < 1 and 0 < x < oo, F(tx) < Ct1+aF(x). reflexive if F is equivalent near zero to a function G for which Lc(0, oo) is reflexive. If F is convex and Lp is reflexive we define F* by F*(t) = sup 0 < a < o o (s£ — ^(s)). Then F* is also a convex Orlicz function satisfying the A2—condition, and LF can be naturally identified with Lp* • Similarly, lF can be naturally identified with lp*. Let us now assume that F(x) = exp(^(logx)) for x > 0 where <j> is a uniformly Lipschitz function; this is, in particular the case when F is convex and satisfies the A2—condition. Then we define Ep to be the closure of the set of functions Ft(x) = F(tx)/F(t), for 0 < t < 1, in C[0,1]; Ep is a compact set in C[0, l ] . We let Cp be the closed convex hull of Ep. It is shown in [6] (see [10]) that IG is isomorphic to a closed subspace of lp if and only if G is equivalent near zero to some G\ G Cp. Lindenstrauss and Tzafriri [8] showed that if G G Ep then IG is isomorphic to a complemented subspace of lp. They asked if the converse was true, i.e. if IG is isomorphic to a complemented subspace of lp, is G equivalent near zero to some G\ G Epl The author gave a counter-example to this with G(x) = xp in [5]. However, these considerations lead Lindenstrauss and Tzafriri to introduce the class of minimal Orlicz functions. We define an Orlicz function satisfying the above conditions to be minimal if F G EG whenever G G Ep. It is easy to see that if F is minimal then it is equivalent to a convex and minimal Orlicz function. In fact if F is equivalent to F and is convex then there exists G G EF which is minimal; further some Fi G EG is equivalent to F by the minimality of F so that Fi is both minimal and convex and equivalent to F. We shall refer to any Orlicz sequence space lp where F is equivalent near zero to a minimal Orlicz function as a minimal Orlicz sequence space. It is shown in [8] that if lp is a minimal Orlicz sequence space and G G Ep then lp « IG and this suggests the conjecture that each such space is prime. In [8] nontrivial examples of minimal Orlicz sequence spaces were constructed. More recently, Hernandez and Rodriguez-Salinas ([3]) gave an explicit example, which actually was introduced for different purposes in [4]. This example, which is reflexive, is given by
(*)
F(t) = t*exp(f;(l - cos(2,r(log*)/2")) n=0
where 1 < p < 00. Let us define a minimal Orlicz sequence space lp to be strongly minimal if, whenever IG is an Orlicz sequence space which is isomorphic to a complemented subspace of lp then G is equivalent to function in Ep. We shall show that there is a non-trivial minimal reflexive Orlicz sequence space which is not strongly minimal and in fact contains a complemented copy of lp for some p\ this space cannot be prime so that the Lindenstrauss-Tzafriri conjecture is false. However, we show that the space lp with F given by (*) is strongly minimal. This does not show that the space is prime; however it does suggest the possibility in view of the following:
Kalton: Minimal and strongly minimal Orlicz sequence spaces
151
THEOREM 1.1. Let tp be a strongly minimal reflexive Orlicz sequence space. Let X be a complemented subspa.ce of tp with an unconditional basis. Then X contains a complemented subspace isomorphic to lp. In particular if X « X X X then X « tp. PROOF: It is easy to show that some subsequence of the unconditional basis is equivalent to the canonical basis of an Orlicz sequence space IG- But then G is equivalent near zero to a function in Ep and so (see [8]) IG & ^F- The last assertion is a well-known form of the Pelczynski decomposition technique." In view of Theorem 1.1 we can relate our example to three open problems (this observation is due to Peter Casazza). If tp is not a new prime space then it either has a complemented subspace X which fails to have an unconditional basis, or lp fails to have the Schroeder-Bernstein Property [l]. This research was initiated during a visit to Spain in May 1989. The author would like to thank Francisco Hernandez for some interesting discussions on this topic and the Universities of Madrid and Zaragoza for their hospitality. 2. Complemented subspaces of Orlicz sequence spaces. We recall that a basic sequence (xn)<^L1 in a Banach space X dominates a basic sequence (y n )£Li m a Banach space Y provided there is a constant M so that for all a i , . . . , a n and n G N ,
HX>y.-ll < M\\ X>*illt=l
t=l
If (xn) is a basis of X then (x*) denotes the biorthogonal functionals in X*. The following lemma is very well-known and we only sketch the proof. LEMMA 2 . 1 . Let X and Y be reflexive Banach spaces with symmetric bases {xn)(^L1 and (yn)£Li> respectively. In order that X be isomorphic to a complemented subspace ofY it is necessary and sufficient that there is an increasing sequence of positive integers (p n )£L 0 w*th Po = 0 and block basic sequences un = ]C?=p n-1 + i aiVii
<M\\y\\\\x*\\ for a suitable constant M independent of m, y, x*. This shows that A is indeed bounded. A very similar argument shows that B is bounded. •
152
Kalton: Minimal and strongly minimal Orlicz sequence spaces
LEMMA 2 . 2 . Let F be a convex Orlicz function such that Lp(O,00) is reflexive (so that < F and F* are both continuous.) Then for 0 < x < 00 we have x < F-1(x)(F*)~1(x) 2x.
PROOF: The right-hand side inequality is obvious from uv < F(u) + F*(v) stituting F(u) = F*(v) = x. For the left-hand inequality, suppose x = note that F(u)/u < F'(u) where F' denotes the left-hand derivative of F*(F'(u)) = uF'(u) - F{u) and (F*)'(F'{u)) < u. Thus (F*)'(x/u) < u F*(x/u) < x. Thus x
upon subF(u) and F. Then and hence
THEOREM 2.3. Let F, G be convex Orlicz functions such that Lp (0,00) and LQ{0, 00) are reflexive. Then for IQ to be isomorphic to a complemented subspace of lp it is necessary and sufficient that there is a constant C and a sequence fin of probability measures each with compact support in (0,1] such that: (1)
I F{F-l{t)x)^j^-
(2)
I F*{{F*)-l{t)x)^^-
< CG{x)
2~n < x < 1
< CG*(x)
2~
n
PROOF: Suppose (l) and (2) hold. Since Ji~ 1 d/i n < oo for each n, we may find an increasing sequence of positive integers (pn)^Lo w ^ h Po = 0 a n ( i a sequence of nonnegative measurable functions (/n)^Lo w ^ n support fn contained in [pn-i>Pn]> s o that fn is nonincreasing on [p n _i,p n ] and such that A(/n > t) = f,t ^ s~1dfj,n(s). Let un(t) = F - ^ / n W ) and let vn(t) = (F*)-1^)). Then clearly
r
F(Un(t))dt = r
Jo
Jo
F*(Vn(t))dt= Jo
Thus ||ttn|| L , = ||»n||L,. = 1.
Suppose a i , . . . ,a n € R with £ t n = 1 G(|a»|) < C" 1 . Let J = {% : 1 < i < n, |a t | > 2" '}. Then 1
1: e * ^ X w h i l e ll£.-*ja.-t*.-|kr ^ Et'Li 2 "*' < I- I* f o l l o w s t h a t the unit vector basis of IQ dominates (un) and a similar argument shows that the unit vector basis of to* dominates (v n). Now let P be the natural averaging projection of Lp onto lp, (which we identify as the subspace of Lp of functions constant on each interval (n — l,n] for n G N) i.e.
Kalton: Minimal and strongly minimal Qrlicz sequence spaces
153
We also use P for the same projection on Lp*. Let un = Pun and vn = Pvn. Since P is bounded on both Lp and Lp*, (un) dominates the block basic sequence (un) and (vn) dominates the sequence (v n ). Now since un and vn are each nonincreasing on [pn-i,Pn] we have Pn
Pn
pi
pi
/
/
Vn(t)dt
fPn
> /
un(t)vn{t)dt
Jpn-1+1
fPn
> /
fn(t)dt Pn-l+l
fn(t)dt.
/
Now we split into two cases. If liminf fpn * fndt < ^ we can pass to a subsequence and apply Lemma 1 to obtain the conclusion. In the other case we can suppose fpn-i*1 fndt - 2 f o r a11 n- I n t h i s c a s e fn(Pn-i + \) > \y and hence u n , P n _ 1 + i > ^ J P - 1 ( ^ ) . Thus (tin) dominates the unit vector basis of lp\ similarly (vn) dominates the unit vector basis of tp* • As (un) and (vn) are dominated respectively by the unit vector bases of LQ and IQ* we conclude that G is equivalent to F in this case. Conversely, let us suppose that IQ is equivalent to a complemented subspace of lp. We denote by (e n ) and (e*) the canonical bases in lp and l*F. We may suppose that there exist normalized block basic sequences (un) in lp and (vn) in lp*, equivalent respectively to the unit vector bases of IQ and la*-, of the form un = X ^ p ^ + i a » e * vn = ]Cp^_!4-i Piei where 0 = p0 < p\ < ... < pn < ... and so that for some 8 > 0, Pn
£
aipi\ > 6.
Note also that
\oiPt\K £
(F(\ai\) + F'(\0i\))<2.
154
Kalton: Minimal and strongly minimal Orlicz sequence spaces
Let K = 4/6 and An = {i : p n _! + 1 < i < pn, max(F(|ai|),F*(|/?t-|)) < K\aipi\}. Let Bn be the complement of An relative to {i : pn-\ -+-1 < i < pn}- Then
Thus if crn = X: iGAn |at-A| then on > f. Now for » G A n we have
so that \Pi\ > (2K)-1(F*)-1(K\ai/3i\). This and a similar inequality for a» imply the existence of a constant c\ > 0 depending only on JP,F* and 5 so that, for i £ An,
Now let un = ^2ieA ot-iti. Plainly the canonical basis of IG dominates (un). Thus there exists a constant C\ so that for any 7 > 0 there exists no = ^0(7) so that if n > no and 7 < x < 1,
Thus for n> no and 7 < x < 1,
and by utilizing the A2—condition, we obtain for a suitable constant Ci depending only F,F* and 6,
Y,
F{xF-1{\aipi\)
A similar argument can be applied to vn = YlieA P*ei m ^F* • ^ v P ass i n g to a subsequence we can for a suitable constant C3 require that for 2~n < x < 1,
^la^.l) < C3G(x)
Now let fin be the probability measure supported on a finite subset of (0, l] given by
5Z l & l
Kalton: Minimal and strongly minimal Orlicz sequence spaces
155
where ea denotes the Dirac measure at a. Then
< 2Cz6~lG{x) as long as 2~ n < x < 1. We also obtain the similar inequality for F* and G* and hence the theorem is proved." Let us now introduce some notation. If lp is a reflexive Orlicz sequence space and 0 < A < oo we shall say that G is A—represented in F if there is a constant C and a sequence vn of probability measures with compact support in (0,1] such that for 2" n < x < 1, A
THEOREM 2.4. Let lp be a reflexive Orlicz sequence space. Then there exist constants 0 < A o < l < A i < o o with the property that for any Orlicz sequence space to to be isomorphic to a complemented subspace of lp it is necessary that G is Ao—represented in F and sufficient that G is Ai—represented in F. PROOF: Clearly it suffices to consider the case when F is convex. Since tp is reflexive, we may assume that LF is reflexive so that there is a constant 0 < a < 1 and a constant c > 0 so that whenever f > 1 and 0 < x < oo, we have F(£x) > c£1+aF(x) and F*{£x) >c£1+otF*(x). For f,z we define A{£,x) = {t : F{t)G(x) > £F(tx)}. Also let B(£,x) = {t : F*(tx) > £F*{t)G*{x)}. We make two claims: CLAIM 1: If 0 < y < 1, and $ > 2 then FiA^G-^y))) C F*(B(ce2-{l+a\
(G*)
CLAIM 2: If 0 < y < 1, and f > 2, then
PROOF OF CLAIM 1: Suppose F " 1 ^ ) e ^ ( ^ G - ^ y ) ) . Then sy > so that F-1(s)G~1(y) < F^isy/Z). By Lemma 2.2, this implies
Thus applying F* to both sides we obtain
Now if we substitute r = ( f * ) - 1 ^ ) and 2 = (G*)~1(y),
CF(F-1(s)G~1(y))
156
Kalton: Minimal and strongly minimal Qrlicz sequence spaces
This implies that r = (F*)" 1 ^) e £(cf a 2-( 1 + a ),{G*)- l {y)) 1
1
PROOF OF CLAIM 2: Suppose (F*)~ (s) e £(£,(G*)~ y).
so that (F*)-1^) 1
1
so F(^F~ (s)G~
< {F*)-1^)^*)-1^).
as required. Then
Hence %£F-1{s)G-1{y) < F^tfsy) and
(y)) < £sy. Now, by the assumptions on F,
and F-X(s) G A(c^ot/21+oc,G~1(y)) as required. We proceed to the proof of the theorem. We pick Ao so that 0 < Xo < a. Suppose first that to ls isomorphic to a complemented subspace of Lp> Then we may choose measures fin as in Theorem 2.3. Let vn = \in o F. Then we have
for 2~n < x < 1.
Next suppose 2~n < ( G * ) - ^ ^ ^ ) ) = y < 1. Then for £ > 2,
Now if Ao < a this leads to an estimate
for 2 n < y < 1 where Cx depends only on C,a and Ao. Passing to a suitable subsequence and combining with (3) gives the result in one direction. For the converse direction, pick Ai so that Aia > 1. This time we suppose vn are given as in the definition of A—representability. Let fin = un o F~l. Then since Ai > 1 it is easy to see that (1) of Theorem 2.3 is satisfied. Now suppose x is such that 2~n < G~1(G*(x)) — y < 1. Suppose £ > 2; then for a suitable constant C independent of n,
C-XiOt
Kalton: Minimal and strongly minimal Qrlicz sequence spaces
This leads to an estimate, for 2
I
n
157
< y < 1,
F*(tx) dfi oF*{t)
for a suitable C\. Changing variables and passing to a suitable subsequence gives (2) of Theorem 2.3.-
3. Examples of minimal Orlicz sequence spaces. We will now describe a method of construction of minimal Orlicz functions suggested by the work of Hernandez and Rodriguez-Salinas [3]. First we fix p > 1. Identify the unit circle T with R/2TTZ. We shall suppose that (fn)^Lo *s a sequence of C1-functions on T (i.e. 2TT—periodic functions on R) satisfying / n (0) = 0 and such that the series £ ^ L 0 2 ~ n ^n converges where Ln = ||/4||oo. For convenience we let Rn = XlfcLn+i 2~kLk so that limn_>oo-Rn = 0. We then define the C1—function <\> on Rby
««)=£;/.
We define for t > 0, F(t) = tp exp(<£(- log t)). We also introduce the functions gn on T defined by gn(0) = Z^=o /fc(2n~fc^). Let us then note that for any u,v we have
\(t>{u-^ v) -
2^
\M^^k—-)-/*(-2*-)l
k=n+l
and hence (4)
\4>{u + v)- <j>(v) - gn(2Hl
+ V)) + 9n(^)\
< 2w\u\Rn.
PROPOSITION 3 . 1 . tF is a minimal reflexive Orlicz sequence space.
PROOF: First, for any e > 0 there exists N — N(e) so that oo
2nRN = 2TT ^2
2 nL
~ n
< e
n=N+l
and hence <j> satisfies an estimate \<)>(u + v) —
158
Kalton: Minimal and strongly minimal Qrlicz sequence spaces
compact subsets of [0,oo). By passing to a subsequence we may suppose that converges in T = R/2?rZ for each fixed n to some an where 0 < an < 2n. Thus 2TTU
\*l>(u) - 9n{~^- + an) + gn{an)\
< 27rjRn|u|.
If we let rn = 2 n (l - (27r)-1an) then |0(w + rn) - i>(Tn) - g,
and hence |^(« + rn) - t/>(Tn) -
This implies that F G EG and so F is minimal." We now impose an additional constraint. PROPOSITION 3.2. Suppose for some increasing sequence of integers Nn we have sup2NnRpfn = A < oo. If F(x) is equivalent to xr for some r then ||0ATJ|OO iS bounded. Furthermore G(x) = xpexp(^(— logx)) is X—represented in F if and only if there is a constant C so that for every n there is a probability measure fin on T such that fexp{\\gNn{$
(5)
+ $o)-gNn[e)-hn[$o)\)diin{0)
0 < O0 < 2TT
wiiere hn(6) = xl>(2N»0/2n). PROOF: Notice first that \
v) - 4>{u) - ip(v)\)dvn(u) < Co for 0 < v < 2 ». If 0 < v < 2 , (4) gives us the estimate /
N
Nn
\
If we define fin on T by / f(6)d^n{d) = / f(27rt/2N»)dvn{t) (5) will follow with C = C0e2*XA.
Conversely we assume (5) we quickly get that for 0 < 0Q < 2n,
1 It then easily follows by a change of variables that G is A—represented in F.m
Kalton: Minimal and strongly minimal Orlicz sequence spaces
159
THEOREM 3 . 3 . Suppose 1 < p < oo. There exists a minimal reflexive Orlicz sequence space tp which is not isomorphic to lp but which contains lv as a complemented subspace. In particular, tp i S n°t prime. PROOF: We pick a sequence of (^-functions {hn)%L0 on T with hn(0) = hn(27v) = 0 and such that if Mn = ||fcn||oo then we have both Mn > 2 n + YllZi M * and
for all n. Let Bn = Halloo- We pick a strictly increasing sequence of integers Nk such that Bk2Nk-l~Nk < 1 for all k, where No = 0. Then define fn by fn = hNk if n = Nk and
2N"RNh = £ j=k+l
j=k+i
<2 so that we can apply Proposition 3.2. Observe first that F cannot be equivalent to any xr since ||flfjvfc||oo > 2*. To complete the proof we show that xp is A—represented in F for every A. Thus Theorem 2.4 will give the result. To show this we estimate by convexity of the exponential function,
However writing gNk = 2"( fc+1 )(0) + £y = 0 2~k' + 1 )(2 J ' + 1 /tf J -) and again using convexity the integral is estimated by
3=0
= 2-(*+D + £ y=o which is bounded, independent of A:, for every A. We can now apply Proposition 3.2. • We now turn to the construction of strongly minimal Orlicz sequence spaces. We recall that lp is strongly minimal if whenever IQ is isomorphic to a complemented subspace of lp then G is equivalent to F. Our example is of the above form with fn(x) = a ( l - cosx) for every n, so that
160
Kalton: Minimal and strongly minimal Orlicz sequence spaces
[3], who also observe that for small enough a, F is actually convex, (it is, of course, always equivalent to a convex function). THEOREM 3 . 4 . Suppose 1 < p < oo and \ct\ > 0. Then if
then F is not equivalent to any xr and tp is a strongly minimal Orlicz sequence space. PROOF: In this case we can apply Proposition 3.2 (take Nn = n.) Thus our proof reduces to analyzing the functions gn{0) — a]L/fc=oU ~~ cos(2fc0)) on T. The fact that F is non-equivalent to any xT is proved in [4], or may be proved from Proposition 3.2 by estimating ||<7n||2- We therefore have only to establish that if G(x) — xp exp(ip(— logx)) is A—represented for some A > 0 in F then G is equivalent to some G\ 6 Ep. This in turn will be achieved by establishing the following result of possibly independent interest. THEOREM 3 . 5 . For any K there is a constant C = C(K) (independent of n) so that if gn(0) = X^=o(l ~~ COS 2*#), fJ> is a probability measure on T and h is a function on T such that for every 0 < 0O < 2n we have \gn(0 + *o) - 9n(0) - h{0o)\2dn{0) < K2 then there exists
forO<0<
2TT.
PROOF: We first introduce the angular distance on T, 6(61,62) = d(0x - 02, 2TTZ). We will be interested in ways of measuring the spread of /x. For 0 < k < n, we define:
=J J =J J In order to estimate these we choose rk for 0 < k < n to minimize s'm2(2k~1(0 - Tk))dfi(0).
It is clear that such a minimizer exists and further if 1 < k < n, we can choose rk from amongst at least 2k possibilities so that if Ak = {0 : 6(2k~l0,2k~1rk) > n/2}
and ak = fj,Ak then ak < | . We further introduce Bk = {0 : 6(2k0,2kTk) > n/4} and bk = fiBk for 0 < k < n. Let Ck = (T \ Ak) n (T \ Bk) and Dk = (T\ Bk) n Ak. Then > 1 — ak — bk and fiDk > ak — bk.
Kalton: Minimal and strongly minimal Qrlicz sequence spaces
161
We next relate ak,bk to a*. Clearly by integrating over Bk we have sin 2 (^)6, < Jsm2(2k-1(e
- Tk))dp{$) < ak.
Thus bk < 10a*. Thus as long as ak < 1/40 we have bk < 1/4. Hence in this case we have fiCk > 1/4. Now if k > 1 and $i e Ck and B2 6 Dk we have 6(2k-1$i,2k~1Tk) < TT/8 but fc 1 A: 1 k 1 k 1 > TT/4 and sin 2(2 fc - 2^ 1 ,2 fc - 2 ^2 ) > <5(2 " ^1,2 - rA;) > 3TT/8. Thus 6(2 - 0u2 - 92) sin2(7r/8) > 1/10. Integrating over Ck x Dk U Dk x Ck gives fi(Ck)fi{Dk) <5ajfc_i. Thus if ak < 1/40 we obtain fiDk < 20ak-i and hence ak < bk + 20ak-i < 20ak-i + 10ak- Since ajt < | we can say in general that for k > 1, ak
<20{ak-i+ak).
Now suppose 0lyO2 G Ck. Then, if fc > 1, we clearly have 6(2k-161,2k-102) k k 2 62). Hence by integration we obtain
- 2r 1 - 2^k~ By induction, we obtain Pi < 2l~nPn + 2TT J
and hence
y=i
< 2TT(1 4- ^ ( 2 0 a y _ i + 30ay)) y=i
y=o
<
162
Kalton: Minimal and strongly minimal Qrlicz sequence spaces
Now returning to the original statement of the theorem we observe that if the hypotheses on \i hold then:
(J j l<7n(<>l + «o) -ff»(l)<M02)) 2 < *K for every 0 < $o < 2ir. Hence integrating again and using Fubini's theorem,
/ / Of* l9n{Bl+6o) ~ 9n{°2+*o)|2?r) ^ w * ) ^4R2However, n
sin(2A: -1(^1 - $2)) sin(2^ 0 + 2 /c - 1 (^ 1 + $2))
9n{Bi + e0) - gn{B2 + Bo) = 2 ^ k=0
so that the integral can be rewritten as <2K2 k=0
or k=o This in turn yields n
/
fik
k=0
It follows that we can pick a G T so that f
n
/ Y, 6i^k^ 2k°W{0) < 2TT(1 + 100K2). J
i._rk fc=O
For this choice of a notice that, for 0 < 0o < 2?r, -0o)-cos2 f c (a + 0o))
< 2TT(1 + 100K2)
k=o and so, for 0 < 0o < 2?r, J \9n{0 + $o) - gn{o + 0Q)\dii{0) < 2TT(1 It now follows easily that IM*o) - [9n{o + ^o) - 9n{°))\ < 4TT(1 + 100K2) + K and the theorem is proved."
Kalton: Minimal and strongly minimal Orlicz sequence spaces
163
PROOF OF THEOREM 3.4, CONTINUED: This is almost immediate. If G(x) = xpexp(i/)(— logo;)) is A—represented in F for some A then setting hn(0) = ^(2 n (^:)) for 0 < 0 < 2TT we certainly obtain that there exists a probability measure fin on T so that 'n(0 + 0o) - 9n(0) ~ hn(e0)\2dtin{6) < K2 where K is independent of n. Thus there exists an with \hn(0)-(gn(e +
*n)-gn(an))\
for 0 < 6 < 2TT where C is independent of n. But then if an = 2n(an/27r) we have that for 0 < u < 2n where C1 is independent of n. It quickly follows that G is equivalent to a function in EF.m References. 1. P.G. Casazza, The Schroeder-Bernstein Property for Banach spaces, Cont. Math. 85 (1989) 61-77. 2. F.L. Hernandez and B. Rodriguez-Salinas, On £p-complemented copies in Orlicz spaces, Israel J. Math. 62 (1988) 37-55. 3. F.L. Hernandez and B. Rodriguez-Salinas, On ^-complemented copies in Orlicz spaces II, Israel J. Math. 68 (1989) 27-55. 4. W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 217, 1979. 5. N.J. Kalton, Orlicz sequence spaces without local convexity, Math. Proc. Cambridge Philos. Soc. 81 (1977) 253-277. 6. K.J. Lindberg, On subspaces of Orlicz sequence spaces, Studia Math. 45 (1973) 119-146. 7. J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971) 379-390. 8. J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces II, Israel J. Math. 11 (1972) 355-379. 9. J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces III, Israel J. Math. 14 (1973) 368-389. 10. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: sequence spaces, Springer, Berlin-Heidelberg-New York 1977.
Type and Cotype in Musielak-Orlicz spaces by A. Kamiriska A. Mickiewicz University and Oakland University
and
B. Turett Oakland University
Abstract. The type and cotype of a Musielak-Orlicz space over a nonatomic measure space are characterized in terms of the Young function that generates the space.
166
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
The notions of type and cotype of Banach spaces were introduced by B. Maurey and G. Pisier [13,14] in the mid-1970's. Since then, these notions have found frequent use in the geometry of Banach spaces [6,9]. In particular, the type and cotype of Orlicz spaces, Lorentz spaces, and general rearrangement invariant Banach function spaces have been investigated [1,2,9,10]. In this paper, the type and cotype of Musielak-Orlicz spaces, a class of function spaces which are, in general, not rearrangement invariant, are characterized in terms of the Young function that generates the space. As a corollary, a characterization of the types and cotypes of Orlicz spaces is obtained. Recall that a Banach space X is of type p for some 1 < p < 2 if, whenever (xn) is a sequence in X satisfying (||x n ||) € ip, then £3nLix'*rn converges almost everywhere on [0, 1]. (Here, and throughout tjhis paper, r n will denote the n-th Rademacher function.) Similarly, a Banach space X is of cotype q for some 2 < q < oo if, whenever (xn) is a sequence in X satisfying 5^^t_i^nrn converges almost everywhere, then (||£ n ||) € ^9. In certain settings, type and cotype are dual notions. Pisier [16] has shown that a i?-convex Banach space is of type p if and only if its dual space is of cotype q where - -+• - = 1. Let us agree on some notation. $ : R
+
(T, S,/x) will denote a measure space and
x T —* R+ will denote a Young function (with parameter); i.e., $ is a non-
negative, extended real-valued function such that, for almost all t, $(0, t) = 0, $(u,t) > 0 for u > 0, and $(u,t) is convex with respect to ti, and such that, for all u > 0, $(u,t) is S-measurable with respect to t. The Musielak-Orlicz space L* is then defined as the set of equivalence classes of measurable functions / : T -> R such that JT $(A:|/(*)|, t) dfi(t) < oo for some k > 0. Under the norm given by ||/||* = inf{r > 0 : JT^(\f(t)\/r,t)dfj,(t)
< 1},
L* is a Banach space. If $ does not depend on the parameter t, i.e., if $(u,t) = $(u) for all t £T, then L* is an Orlicz space. More information about Musielak-Orlicz spaces can be found in [7], [15]. In this paper, several notions restricting the growth of the Young function $ will be considered. The most classical notion is that of a A2 -condition. A Young function $
Kaminska & Turett: Type and cotype in Musielak-Qrlicz spaces
167
satisfies a A2- condition if there exists a positive constant K and a nonnegative integrable function h such that $(2u,t) < K$(u,t) + h(t) for all u > 0 and almost all* G T. Growth conditions intimately connected to the A2-condition will now be defined. Definition 1. A Young function $ satisfies condition Aq (q > 1) if there exists K > 0 and a nonnegative integrable function h such that $(Au,£) < K\q($(u,t)
-f h(tj) for
A > 1, u > 0, and almost all t G T. A Young function $ satisfies condition A*p (p > 1) if there exists If > 0 and a nonnegative integrable /i such that $(\u,t) > KXP ($(u,t) — h(t)) for A > 1, u > 0, and almost all t. Recall that, if $1 and $2 are Young functions, $1 is said to be equivalent to $2> denoted by $! ~ <£2, if there exist positive constants K\ and K2 and nonnegative integrable functions /&i and h2 satisfying $\(Kiu,t)
< $2(w,t) + hi(t) and $2(^2^,*) <
$i(u,t) + h2(t) for all u > 0 and almost all t 6 T. It is easy to check that conditions Aq and A*p are preserved under equivalence and that a Young function satisfying condition Aq also satisfies a A2-condition. The following lemmas collect some equivalent formulations of the conditions Aq and A*p. In order to state the next lemma, a generalization of the notion of a Young function with parameter is needed. An Orlicz function (with parameter) is a nonnegative, extended real-valued function $ : R+ x T -» R+ such that, for almost all t, $(0,t) = 0, $(u,t) is nondecreasing and continuous in w, and $(u,<) tends to 00 as u tends to 00, and such that, for all u > 0, $(u,t) is S-measurable with respect to t. The notion of equivalence will be used between a Young function and an Orlicz function. Some of the ideas in the next lemma can be found in [12]. Lemma 2. Let $ be a Young function and q > 1. The following assertions are equivalent: (a)
$ satisfies condition A9.
168
(b)
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
There exist K > 0 and a nonnegative measurable function g such that f$(g(t),t)dt
< oo and, for almost all t e T, $(Au,*) < K\«$(u,t)
for
all A > 1 and u > g{t). (c)
There exists an Orlicz function $ equivalent to $ such that, for almost all t e T, $(\u,t)
(d)
< \
There exists an Orlicz function $ equivalent to $ such that, for almost all t GT, ${v}lq,t)
is concave in u.
Proof. The proof that (a) implies (b) is straightforward. Indeed, with h as given in the definition of condition A 9 , define g to be the (measurable) function satisfying
Assume that (b) is satisfied and define
if
v
1 $(v,t) Then ip is nonincreasing and continuous and, if v > g(t), —-— Define $(u,t)
=
JQM
1
ip(v^v*"
$(v,t) < (p(y,t) < —
.
dv. $ is an Orlicz function (but perhaps not a Young
function). To check that $ is equivalent to $, note that if u > g(t), $(u,t)
>
$(u,t)IKq > $(u/Kq,t). Thus, for all u > 0, $(u/Kq,t) < $(u,t) + $(g(t\t). Since ^(.9 (')> ') ^ £*> t n ^ s ^s n a ^ °^ what is needed. For the other inequality, note
g(t)
<*(u,t) Again, since $(jf(-), •) 6 I 1 , $ ~ $.
Kaminska & Turett: Type and cotype in Musielak-Qrlicz spaces
169
Next, an easy change of variables and the nonincreasing nature of (p yields that u,t) < \
is concave in u.
Finally, to show that (d) implies (c), note that concavity implies that, for A > 1 and
/«, t) < $C«^V0 AU
u>0.
Thug 4(Al/ful/f>
t)
<
A # ( u l / f > t) f o r A
> !
md
U
This completes the proof of Lemma 2. There are analagous characterizations of condition A*p. Lemma 3. Let $ be a Young function and p > 1. The following assertions are
equivalent: (a)
$ satisfies condition A* p .
(b)
There exist K > 0 and a nonnegative measurable function g such that f$(g(t),t)dt
< oo and, for almost all t € T, $(Au,<) > KX^^(u,t)
for
all A > 1 and u > g(t). (c)
There exists a Young function § equivalent to $ such that, for almost all t e T, $(Au,*) > \P$(u,t)
(d)
for u > 0 and A > 1.
There exists a Young function $ equivalent to $ such that, for almost all t G T, ^ u 1 / * , * ) is convex in u. The proof of this lemma is analogous to the proof of Lemma 2 where, in (b)
implies (c),
I suPff(<)<«<«
>' ^
if t; > g(t).
In this case, note that $ is a Young function. The connections between the A2-condition, condition A g , and condition A*p are given below. The conjugate function of a Young function $ is defined by
170
Kaminska & Turett: Type and cotype in Musielak-Qrlicz spaces
$*(v,t) = sup{uv — $(u,£) : u > 0}. It is easy to check that, for a Young function $, ($*)* = $ and that, if two Orlicz functions are equivalent, so are their conjugates. Proposition 4. The following assertions are equivalent: (a)
$ satisfies a A2-condition.
(b)
$ satisfies condition Aq for some q>l.
(c)
$* satisfies condition A*p for some p > 1.
In fact, if p, q > 1 satisfy - + - = 1, $ satisfies condition A 9 if and only if $* satisfies condition A*p. Proof. It is clear that (b) implies (a). That condition (a) implies condition (b) follows from the proofs of Theorem 4.1 in [8, p. 24] and Lemma 3.1 in [11, pp. 21-22] and the fact that if $ satisfies a A2-condition, there exists an equivalent $ satisfying $(2u,t) < K$(u,t) for some K > 0, u > g(t) where $(#(•),•) is integrable, and almost all t G T. The equivalence of conditions (b) and (c) follow from conjugate duality. Indeed, assume condition (b) is satisfied; so, by virtue of Lemma 2, there is no loss of generality in assuming that $(Au,t) <
K\q$(u,t)
for all A > 1, u > 0, and almost all t. Taking the conjugate of each side yields $*(v/\,t)
> KX^*(v/KX^t)
for all A > 1, v > 0, and almost all t. With
1
x = v/KX* and /i = A'A*" , it follows that $*(/iz,*) > K1^^)pP$*(x,t)
which
completes the proof since the condition A*p is preserved under equivalence. With these preliminaries on growth conditions behind us, their connections to the type and cotype of Musielak- Orlicz spaces can be stated. Theorem 5. Let (T, E, /x) be nonatomic and let $ be a Young function with parameter. Let 1 < p < 2 < q < oo. a.
The Musielak-Orlicz space L* is of cotype q if and only if $ satisfies
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
171
condition Aq. b.
The Musielak-Orlicz space L* is of type p if and only if $ satisfies a A2-condition and condition A*p.
The sufficiency in (a) is proven in a manner similar to the proof of Theorem 2 in [2] by Z. G. Gorgadze and V.I. Tarieladze; the extra ingredients needed are contained in Lemma 2. In order to prove the necessity in (a), a few technical lemmas will be used. Lemma 6. Let (T, S,/x) be a nonatomic measure space. For natural numbers m,n, and j , let fnj:T fnj
—* R"1" be measurable functions satisfying fnj+i
<
and JT sup fnj dfj. = oo. Then there exists a sequence (An) of pairwise disn>m
joint sets in E and a strictly increasing sequence (£j)j>o of nonnegative integers with ^o = 0 such that Yln=t _i+i I A fni ^ = 1 f° r e a c ^ 3' € N. Proof. By the decreasing nature on (fn,j)j, either sup fnj is almost everyn>l
where finitely-valued for large j or, for each j , sup / n j - takes on the value -foo n>l
on a set of positive measure. In the first case, assume, without loss of generality, that sup fnj
is everywhere finitely-valued for all j £ N. The nonatomic na-
n>l
ture of the measure spaces gives rise to CJ G S such that Jc,1 sup / n>1 d\i = n >! 2. Since (max / n l) increases to sup / n i , there exists £\ G N such that !<"<*
1 < fc, i
Jc
'
max
n>l
/ n> i c?^ < 2.
Choose a measurable set C\ C C{ with
1 ^.n^.ti
max fnii dfi = 1. For n = 1,...,^!, defined = {t G Ci : max fkti(t)
/ n > i(0 > /.•(*) for » = l , . . . , n - l } . Then ^ Since J c /T\CI
SU
1
sup / n 2 c?/i < n>*! + l
n 2
P / » ^A*
=
fc ea
X
sup fnidfi
=
/ n>1 = ^ ' = 1 /B>1 x ^ . and =
1, it follows that
n>l
°°* R^P ting the above argument yields £2 > 0, and
n>li + l
sets A^+i,... Ai2 with the desired property. The remainder of the proof, in this case, follows similarly.
172
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces In the second case, assume that, for every natural number j , Dj is the set of
positive measure on which sup fnj takes on the value + oo. Then the sequence (Dj) decreases to a set D. Depending on whether fiD is positive or zero, the nonatomic nature of \i or disjunctification yields a sequence (Bk) of disjoint sets on which sup fn k(t) = oo for t G Bk. An argument similar to the first case now n>l
completes the proof.
The next lemma provides an equivalent formulation of condition Aq. The definition says that the (nonnegative) function
HK{1>)
= sup { ——- $(\u,i) — u>0
*> XV A ^
§(u,t) > should be integrable for some K > 0. Here it is noted that one need only consider A that are large powers of 2.
Lemma 7. The Young function $ satisfies condition Ag if and only if $ satisfies a A2-condition and there exists K > 0 and a natural number m such that / sup sup {
$(2nu,t) - *(u,t)\ dfi< oo.
n n>m n>m u>0 u>0 l> A • Z 9
J
Proof. In order to conserve space, we shall denote the above integrand by hK,m(t)' Since the necessity is clear, assume $ satisfies a A2-condition and fhK,m dfi < oo. Let Kx > 2 and 0 < hx e L1 satisfy $(2u,t) < for u > 0 and t G T and let t > max{iC1m,il'iv'1}. Then
Ht(t)=
sup sup | J L $ ( A M ) -*(«,*)} l
u>0 ^ A »
+ sup sup {— >
^A
)
Kaminska & Turett: Type and cotype in Musielak-Qrlicz spaces
173
Consider the first term: sup
sup { —-$(Au,
l
{
z/'m
J{m
1
In order to bound the second term, let A > 2 m and choose n 6 N such that
2"< A<2" +1 . Then
sup
^
SUp
u>0
Combining the above bounds yields that Ht(t) < /iK,m(0 + 2/ii(t) for t £ T. Therefore Hi E L1 and the proof is complete. We are now prepared to prove the remainder of Theorem 5. Proof of the necessity in Theorem 5a. Assume that L* has finite cotype q but that $ fails to satisfy condition A g . The finite cotype of L* implies that $ satisfies a A2 -condition since otherwise L* would contain an isometric copy of £°° [4]. Thus, by Lemma 7 and the notation defined in its proof, for all K > 0 and all natural numbers m, J h,K,m dfi = 00. In particular, for all natural numbers m and j , / sup sup < -—r-— $(2 n u,t) — $(u,t)i dp = 00. Let n>m u>0 ^2 J 2 '
)
174
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
{ui : i G N} denote the set of nonnegative rational numbers with ui — 0; taking the supremum over the nonnegative reals or over {w,} in the preceding integral yields the same result. Now define Anij = {t G T : ^r^T $(2nu»,*) > $(tij,*)} and xnj(t)
= sup m XAnij(i)> Then / sup — - $(2nxnj(t),
t) dp = oo for all
n>m £
i
m , j in N. By the definition of Anij, An)i^+\ C AHfij and hence xHj ;_|_i < Finally, defining fnj (t) = ^ ? $(2nxnj(t),t)
xnj.
yields a double sequence (fnj) of
measurable functions on T such that fn,j+i < fnj and / sup fnj dp = oo for n>m
all natural numbers m and j . If we knew that the fnj 's assumed only real values (almost everywhere and for sufficiently large n and j), Lemma 6 would apply. So, assume momentarily that there exist increasing sequences (n*) and (jk) of natural numbers and a set Ck G E with pCk > 0 such that fnk,jk(t)
— oo for
all t G Ck. The (Ck) may be chosen to be pairwise disjoint. Then, for all t G Cky lim $(2 n * max wt- XAn
,• ^ (^), 0 = °o- Choose £k G N and a measurable
subset Dk of Cjk such that j
D
^ r
^ ( 2 n * max w»XAnjfc
ijk
(t)->t)dp
=
Define gk = max^ WiX^nfc,.,ifcXDfc; then, by the definition of Anjk>ljJfc, nk
k
J 7fjrj-j$(2 gk(t),t) dp < 2* f $(gk(t),t)dp. JD $(2nkgk(t),t)dp
Thus § $(gk(i),i)dp
1. 1 =
J
= 2~ * and
= 2nkq. Let T1? •••, T^^jt^j be pairwise disjoint subsets of
Dk such that / T . $(2 n *^(t),^)c?/i = 1 for i = 1, • • •, f2n*«]. Then | | ^ XT.Ik = 2" n * for i = 1, • • •, If2n*9 1 and
00
I
2 1
EE
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
175
Moreover
*=1
J
< oo. Since $ is an even function, when the functions gkXTi a r e ordered in the order that they are summed above, the computations yield a sequence (zn) in L* such that J^ n znrn converges almost everywhere but ^ Iknll9 = °°- This contradicts that L* has cotype q. We may therefore assume that fnj is real-valued for each n,j in N. Lemma 6 then implies that there exists a sequence (Ak) of pairwise disjoint sets in S and a sequence (£n)%L0 w i t h 4> = 0 such that
^
J
-—j$(2nk
xki(t), t) dfi = 1.
For eachfixedt, consider the set Si = {k G [A-i + l, ^,] : JAk $(2nkxki(t),t) 1}. Then ^
/
nk
——$(2 x ki (t),t)dn
dfi <
< - . Therefore, by eliminating the k in
»%, there exists a (new) sequence (Ak) and a (new) sequence (^n)£°_0 with ^o = 0 such that, for each i e N,
and, for each k = l{-\ + 1,
Let mjt = [ / A $(2n*Xjbj(t),t)c?/iJ and choose pairwise disjoint measurable subsets Tj*, • • •, T£k of A* such that JTft $(2nkxki(t), t) dp = 1 for j = 1, • • •, m*.
176
Kaminska & Turett: Type and cotype in Musielak-Qrlicz spaces
Hence, for j = 1, • • •, mk, \\xkiXr> II* = 2 ~ n *oo
E 1
li
E Jk^
mk
oo
E i*« XT; ik = E j l
1
Then
li
E
2 n
~ " •»*
/ b / + l
t=l oo
«=1 =
CO.
On the other hand, /.
OO
li
rn
k
/ #(V y y t=i t=/,_,+i j=i
[t),t)
•>
li
OO
E =E t=i A
oo
^E E
= E 2 " ( E 2~n*g oo
t=i <
CO.
As in the preceding case, this contradicts that L* has finite cotype q and the proof of Theorem 5a is complete. Part (b) of Theorem 5 follows from part (a) via duality and the following facts: the result of Pisier mentioned in the introduction; a result of Pisier [17]
Kaminska & Turett: Type and cotype in Musielak-Qrlicz spaces
177
which states that a Banach space has type p > 1 if and only if it does not contain ^ ' s uniformly, i.e., if and only if the Banach space is U-convex; the result that, when $ satisfies a A2-condition, (X*)* = £** (see [7] or [15]); and the fact that, in a Musielak-Orlicz space, reflexivity and i?-convexity are equivalent [5,6] and can be characterized by both the Young function $ and its conjugate function satisfying a A2-condition. As an immediate corollary of Theorem 5, Proposition 4, and the result of Maurey and Pisier [13] that a Banach space has cotype q < oo if and only if it does not contain £^s uniformly, we obtain: Corollary 8. A Musielak-Orlicz space Z* does not contain ^ ° ' s uniformly if and only if $ satisfies a A2-condition. An analagous characterization of when L* contains ^ ' s uniformly is already known (see [5],[6]). It should be noted that the proof of Theorem 5, although cumbersome in the setting of Musielak-Orlicz spaces, simplifies significantly if one only considers Orlicz spaces. In fact, in the Orlicz space setting, we have the following: Corollary 9. Let L* be an Orlicz space over a nonatomic measure space and l < p < 2 < g < o o . If /zT < 00, then: a.
Ir* has cotype q if and only if there exists K > 0 and u0 > 0 such that $(\u) < K\i$(u)
b.
for all A > 1, and u > u0.
X* has type p if and only if $ satisfies a A2-condition and there exists K > 0 and u0 > 0 such that $(Au) > K\P$(U)
for all A > 1,
and u > UQ. If fiT = 00, the the above inequalities have to hold for all u > 0. The above characterization of cotype of an Orlicz space has been done pre-
178
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
viously in [9] and [10] and the characterization of type was done in [9] under the assumption of a A2-condition. In fact, in [10], Maleev and Troyanski define a modulus on arbitrary Banach lattices and employ this modulus to characterize the cotype of Banach lattices. In particular, they compute the cotype of L* when T is (0, oo), [0, 1], or [1, oo). Additionally, in the Orlicz space case, these results can be stated in terms of indices. If pT < oo, define lower and upper indices a* = sup {p : iof A,u>l
. \."\
> 0}
\P$(u)
$(Au) oo}; E fiT = oo, the supremum and infimum are taken over all positive u and all A > 1. For information on indices and their applications, see [9] or [11]. The following are essentially known and rephrases our results in the Orlicz space case in terms of indices. Corollary 10. Let (T, S, p) be a nonatomic measure space and let $ be a Young function (without parameter). a.
The Orlicz space £ * has finite cotype if and only if fi$ < oo.
b.
If q > max{/?$,2}, the Orlicz space L* has cotype q.
c.
If the Orlicz space L* is of cotype q, then q > max{/?$, 2}.
d.
The Orlicz space L* has type larger than 1 if and only if 1 < a $ < P* < o o .
e.
If /?$ < oo and 1 < p < min{a$,2}, then the Orlicz space L* is of type p.
f.
If the Orlicz space X* is of type p > 1, then ft* < oo and p < min{a$,2}.
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces
179
We note that the main result can be aplied to the Nakano space; i.e., the Musielak-Orlicz space with $(tt,t) = upW and the function p : T —> [l,oo) measurable. In fact, the Nakano space has cotype q exactly when p(t) < q for almost all t G T and it has type p exactly when p(t) > p for almost all t E T. We conclude with some examples: 1. For any p € [1,2] and q £ [2, oo), there exists an Orlicz spaces of type p and cotype q. Indeed, it suffices to take JL* = IS C\Lq where $(u) = max(u p , uq). 2. An Orlicz space need not have its upper index as a cotype. Let q > 2 and define if tx = 0
0
Then /3$ = q, but sup
= oo. Thus L* has cotype q -f e for every e, but
u>0
does not have cotype q. In closing, we want to thank Professor Troyanski for informing us of his and Professor Maleev's results.
REFERENCES
1.
J. Creekmore, Type and cotype in Lorentz Lpq spaces, Proc. Acad. Amsterdam, A 84 (1981), 145-152.
2.
Z. G. Gorgadze and V.I. Tarieladze, On geometry of Orlicz spaces, Probability Theory on Vector Spaces, Lecture Notes in Mathematics, Volume 828, Springer-Verlag, 47-51.
3.
J. Hoffmann-j0rgensen, Probability in Bspaces, Lecture Note Series, No. 48, Aarhus Univ., 1977.
4.
H. Hudzik, On some equivalent conditions in Musielak-Orlicz spaces, Commentationes Math. 24 (1984), 57-64.
5.
H. Hudzik and A. Kaminska, On uniformly convexifiable and B-convex Musielak-Orlicz spaces. Commentationes Math. 25 (1985), 59-75.
180
6.
Kaminska & Turett: Type and cotype in Musielak-Orlicz spaces H. Hudzik, A. Kaminska, and W. Kurc, Uniformly non-£ 1 (n) Musielak-Orlicz spaces, Bull. Acad. Polon. Sci. 35 (1987), 441-448.
7.
A. Kozek, Orlicz spaces of functions with values in Banach spaces, Commentationes Math. 19 (1976), 259-288.
8.
M. A. Krasnasel'skii and Ya.
B. Rutickii, Convex Functions and Orlicz
Spaces, P. Noordhoff, Ltd., Groningen, 1961. 9.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer-Verlag, 1979.
10.
R.P. Maleev and S. L. Troyanski, On cotypes of Banach lattices, Constructive Function Theory '81, Sofia, 1983, 429-441.
11.
L. Maligranda, Indices and Interpolation, Dissertationes
Mathematicae
(Rozprawy Mathematyczne), No. 234, Warsaw, 1985. 12.
W. Matuszewska and W. Orlicz, On certain properties of ^-functions, Bull. Acad. Polon. Sci. 8 (1960), 439-443.
13.
B. Maurey and G. Pisier, Caracterisation d'une classe d'espace de Banach par des proprietes de series aleatoires vectorielles, C. R. Acad. Sci.
Paris
277, Series A(1973), 687-690. 14.
B. Maurey and G. Pisier, Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach, Studia Math. 58 (1976), 45-90.
15.
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Volume 1038, Springer-Verlag, 1983.
16.
G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Annals of Math. 115 (1982), 375-392.
17.
G. Pisier, Sur les espaces de Banach qui ne contiennent pas uniformement de i}n, C. R. Acad. Sci. Paris 277, Series A(1973), 991-994.
ON THE COMPLEX GROTHENDIECK CONSTANT IN THE n-DIMENSIONAL CASE
Hermann Konig (Kiel)
Abstract. We derive some new bounds for the complex Grothendieck constants K G C (n) for n-dimensional Hilbert spaces. One has e.g. 1.152 < KGC (2) < 1.216. For n > oo the estimates yield Haagerup's bound K G < 1.405.
1.
Introduction Grothendieck's inequality is the following well-known CLPD.
Theorem 1. For any n > 2 there is K G (n) > 0 with the following r
Given any finite matrix (aj k )j t k =i
and given any vectors
(x^)-^
property:
with
, (yk^k^i
m
an
^-dimensional
Hilbert
space
H n , one has Z
a jk < x : , y k > | < K G (n)
j,k = 1
J
J
sup
|| x, || || y k
1< j , k < m
J
We use superscripts IR and C to indicate the different values in the real and complex cases; the complex constants are smaller than the real ones. Clearly, the sequence ( K G ( n ) ) n > 2 is increasing in either case. The best known upper estimates for the Grothendieck constant
K G : = lim K G (n)
are due to
Krivine and Haagerup,
K
GC
2 —r^-r
= 1.40S
[Ha] .
Here k o is the solution of a certain equation involving elliptic integrals, see
182
Konig: Complex Grothendieck constant in the n-dimensional case
below. The upper estimate for KG
is conjectured to be the exact value by
Krivine, whereas the bound for KG is not considered to be precise by Haagerup. We remark that Pisier CPi] first proved that KGC < KGR by showing K
Any continuous linear operator T : lt
> \£ is 1-summing with
b) Any continuous linear operator T : 1^
> lt is 2-summing. If the 2-summing
norm is calculated only on n vectors, one has 7r2(n) (T) < K G (n) || T || . c)
For any operator between Banach lattices, T: X
> Y, and any sequence
x t , • • • x n € X, one has II ( |
I Txj | 2 ) 1 / 2 || < K G (n) || T || || ( i | i | Xi | 2 ) 1 / 2 ||
To state the estimate for K G C (n), we define
> [-1,1] by
7i:/2
cos2a
cp(s) = s f - da , s€[-l,l] . O Vl-s 2 sin 2 a By Haagerup [Ha],
=
(32k+t t 2k
f
+1
,
te[-l,n
k=O
with (3j = 4/TU and p2k + i - 0 for k > 1. Formal calculations yield o
_
8
ft_nR_
1 6
o
_
8
O
R
_
4 8 O
n
_
3 1 3 6
Konig: Complex Grothendieck constant in the n-dimensional case
Theorem 2. Let
solution
A2k + t (n) =
n
^or
^ o , n > 2.
There
is a
unique
s n of the equation 00
2
2k+ 1
Z A 2 k + i (n) k=o
P 2k + i
For this value s n , K G C ( n ) < —(*s
As n
k €
183
> oo , A 2 k + 1 ( n )
> 3k
o
)
= 1 + sn , 0 < sn < 1 .
. The sequence
(—^—>) n > 2 As
increasing.
. Thus t h e limiting equation is
2 p t
9<s n )
i.e.
< —r^—r =
= -^ (l + s o ) . ,1>s—j-
= 1.4049 . He conjectures
t h a t this might be slightly improved t o n/2
KGC < |cp(i) T 1 = ( J o (extending KG
R
<
2ln
c o
d a ) ' 1 £ 1.4046
J^
~ I *P^*^ I~ f ° r
tn
e corresponding function cp(s) =— sin" s
in the real case. A corresponding conjecture for a slight improvement of the bound for K G c (n) of the theorem is the Conjecture. Let B 2 k M ( n ) = <-1> k | n l » - • • ( " : < £ - ? ) )
for
•
n
~ 2"
Then there is a unique solution 0 < c n < 1 of
and K^
(n) < -£— . The sequence
(^~)n>2
ys
increasing.
This would be a formal analogue of the r e s u l t s of Krivine CKr] for K G K (n) and would give KG (3) < 4/TU, in particular.
We
make make a few
comments
in
section 3 on t h e origin of t h e formula in the conjecture.
The equations in the theorem and the conjecture can be solved numerically by using tables for the complete elliptic integrals E and K and noting that 9<s) = £ ( E ( s ) - ( 1 - s 2 ) K ( s ) ) .
184
Konig: Complex Giothendieck constant in the n-dimensional case
Lower numerical bounds for KQ C (n) can be given by using the (unpublished) method of Davie who showed that KG^ > 1.677
and
KGC > 1.338.
Some remarks on this are found in section 4. The following table gives numerical values for the bounds on K G (n), rounded to 4 decimal digits, for some values of n. The upper and lower bounds differ by about 5 %.
n
K G C (n) >
K G C (n) <
2 3
1.1526 1.2108 1.2413 1.2600 1.2984 1.3181 1.3300 1.3381
1.2157 1.2744 1.3048 1.3236 1.3628 1.3834 1.3962 1.4049
4 5 10 20 50 oo
2.
Conj. K G c ( n ) < 1.2146 1.2732
(4/TC)
1.3036 1.3224 1.3618 1.3827 1.3957 1.4046
The addition formula for complex spherical harmonics The proof of the theorem relies in an essential way on the addition formula
for sperical functions. To define complex spherical harmonics in n variables, fix n€lN and choose k€lN o . Let m = Ck/2] and 1 = k - m , i.e. 1 = m if k is even and I = m + 1 if k is odd. Let H k be a complex polynomial in the 2n variables z1,-zn
, Zj.-'-z,, which is harmonic and homogeneous of degrees (l,m), i.e.
which satisfies ^
Hk(z,z) = 0
(b) H k (Xz,(iz) = X l t i m H k ( z , z ) where z = (z1? •• z n ) , z -(zx,---ln). S n " 1 (C), define Y k : S n ~ 1 (C)
,
X,[i£€
On the complex n-dimensional sphere > C by
, C € S n " 1 (C) .
Konig: Complex Grothendieck constant in the n-dimensional case
185
This "restriction" of H k to S n *(C) is called a spherical function of order k in n dimensions. The set of all such spherical functions is denoted by lMJ^ = %fn k\ we fix n and often omit this index. Let N k := d i m c 55^. [One has [DGS] /n+k-1\
M
N
/n +l - n
/n+k-2\
/n+l-2\
k = 1 n-1 j 1 n-1 J - ( n-1 J ( n-1 J
For a , 3 > - 1 , denote by P k
a
\
J•
the Jacobi polynomial of degree k on IR, nor-
malized by P k ( o c ' 3 ) ( l ) = 1. Thus the ( P k ( o c ' 3 ) ) k e l N o are (the) nomials
on
[-1,1] with respect
to
the
weight
orthogonal
(l-t)a
function
(1 + t) .
polyFor
t € IR, let (2t2-l) qk(U
The ( q k ) k € N o for z € C ,
are
:=
k even
I t P^n-^O
orthogonal on [-1,1] with respect to |t| ( l - t 2 ) n ~ 2 .
Further,
put f
k even
q q k(lzl)
^ I it\ For the purpose of surface integration, identify S n
(C) = S
by dw = do)n the (rotation invariant) surface measure on S by
J
dw n = 1 . Consider ^ k = ^
n
(IR) and denote
n
(C), normalized
as a subspace of L2= L 2 ( S n " 1 (C) ,du n )
n k
and denote by (Yk j)j = 1 an arbitrary but fixed orthonormal basis of ^fk in L 2 . The following result is stated by Delsarte-Goethals-Seidel [DGS] who refer to Koornwinder [KoL
A d d i t i o n t h e o r e m f o r s p h e r i c a l f u n c t i o n s . For N
all
k € I N O and
x,y€S
n l
(C)
k
Y kj (y) .
j— ^
In the real case, a corresponding formula holds, if the polynomial q k is taken to be P k ( ~ ^ ~ ' ~ 2 ~ ) , cf. Miiller [Mlil
Corollary.
For
N
all
k
k € IN O
J s n-l
Qk<
and
< x
x,y€Sn
' > >> Q k <
N
*y >)
dun(y)
=
Qk(<x,z>)
186
Konig: Complex Grothendieck constant in the n-dimensional case
Proof of the corollary. The left side equals by the addition theorem Nk
^-
k
£
Y kl (x) Ykj(z)
i ,j — 1
N
,
J
Sn
»
Yki(y)
Ykj(y) dcon(y)
k
= 75- . 2
Ykj(x)
Ykj(z)
= Qk(<x,z>)
.
Nk
Thus Nk Q k ( < • , • > ) =
2 Ykj <8> Ykj is the kernel of the orthogonal projection
n 1
in L 2 (S ~ ,dw) onto ^ n > k (as an integral operator). The addition theorem may be useful for other purposes in the geometry of Banach spaces. No simple, explicit proof is given in CDGS], CKo] (in particular, not for the case of "k odd" which we need). We thus supply a sketch of a fairly simple proof obtained by modifying the arguments of MUller CMU] in the real case.
Sketch of proof. (i) The function F k : S n ~ 1 ( C ) x S n " 1 ( C )
>€ given by
F k (x,y) = X Y k j (x) Y k ,(y) is unitarily invariant: For any unitary (nxn)-matrix A, F k (Ax,Ay) = F k (x,y). To see this, note that x •
>Yk:(Ax) is a spherical function in ^ k , too Nk
(for all j). Basis expansion gives Y k j(Ax) = Z C = (cj r )j
r=1
cjr
Y k r (x).
The matrix
is unitary for all k€ INO since by the orthogonality of the
Ykj and invariance under A s
ii =
f Y k i (x) Y kj (x) dw(x) s n'-l
Nk
I Yki(Ax) Yki(Ax) du(x) = S sn^l
r=l
c> cirk . J
Using this, direct calculation yields F k (Ax,Ay) = F k ( x , y ) .
(ii) Let x , y € S n ^ C ) . Choose A € U ( n ) :
unitary with Ay = e n = ( 0 , - • 0,1). Let
= Ax = T) e n + / l - l T ) l 2 7]n~l with 7)€C , rj"" 1 € S n ~ 2 ( C ) . Then
Konig: Complex Grothendieck constant in the n-dimensional case
187
< x , y > = < A x , A y > = < £ , e n > = T) , F k ( x , y ) = F k ( £ , e n ) . Since G:= ( B e U ( u ) I B e n = e n } * U ( n - l ) , for any two i\n~l,9)n"1
€ Sn~2(C)
there is B € G with BT)"" 1 = 9f n - 1 . Thus F ( £ , e n ) is independent of the special choice of T]"" 1 € S n ~ 2 ( C ) . Hence with C = 7)e n + J\-\T\\2
en_t
F k ( x , y ) = F k ( C , e n ) = 0 k (r)) = ® k ( < x , y > ) for some suitable function O k : { z € C | Izl < 1 }
...... , . , (m)We claim that
_ . . r 0. (lr)l) * k <7,) = { ^ ^ ^
>C.
k even |T||)
R
odd
with sgn 7] = T)/ IT)I : By definition F k ( • , e n ) € ^ k . Thus with m = [k/23,1 = k - m there is a (1 , m ) - h o m o g e n e o u s polynomial H k with F k ( £ , e n ) = H^ ( £ , £ ) , £ as
in
(ii).
Thus for
all
9€[0,2TT:)
O k (e i e T]) = F k ( e i e C , e n ) = H k ( e i B £ , e ~ i G
O
= ( e 1 0 ) 1 ( e - i 0 ) m H k ( C ? ) = ( e i Q ) E 0^7)) with s = 0 if k is even and £ = 1 if k is odd.
(iv)We claim that ® k is a polynomial of degree k of the form r cpm(t2) 11
k
k even k odd
m
with some polynomial cpm of degree m = Ck/23: By the (1 , m ) - h o m o g e n e i t y of the function H k of (iii) . _ _ . H k ( Uj , • • • U n , U^ , • • • , U n ) = ¥T
I ZJ
"? ZJ
. _ _ , i _J ^ l - i m - i' U 1 ' * ' * u n - l U 1 »' " ' u n - 1 ' u n u n
where A j _ i ) f n _ j is ( l - i , m - j ) - h o m o g e n o u s in the first pair of (n-1) variables. Let ^ = r)e n + y 1-IT)I 2 T\n~ n
n
as above. Since F k ( £ , e n ) = H k (C,^) is indepen-
2
dent of 7] ~ e S ~ ( C ) , the coefficient functions A,_j Iu,| 2
+•
•+ | u n _ t | 2 = U , I J 7 + - - - + u n _ ,
un_,
m_j
only do depend on
.
Being ( l - i , m - j ) - h o m o g e n e o u s polynomials, they are of the form
188
Konig: Complex Grothendieck constant in the n-dimensional case
,
, ,
k
A i _ i > m _ j ( U | , • • - U J J . J , u l f - • - u n _ j ) = Cy with
k
~(^
f )
€ INO. C o n s i d e r £ = t e
n
+ yi-t
2
,?
2v
( l u t r + • ••+ lun_tl e n _ j for t € [ - l , l ] .
k-(i-» o—
)
^
Then
O k (t) = F k (C,e n ) = Hk(C,T> 1
m i= O
d-t 2
enJ
where q k * 0 only if
k
^+V
€IN O . If k is even (odd), thus (i + j) has
to be even ( o d d ) . Hence
> IR is an integrable function and e € S n MC), introduction of
(v) If f : [0,1]
(2n-l) polar coordinates in S n " 1 (C) = S 2 ""^^) shows that (independently of e) 1
f(|< x , e > | ) d o ) n ( x ) =2(n-l) J f(t) t ( l - t 2 ) n " 2 d t .
J S n ~ 1 (C)
O
(vi)We found for x,y€S n " 1 (C) and m = Ck/2] that Fk(x,y) = O k ( < x , y > )
f Pm(lx'y>l2) = j <x ,y > 9 r n ( I <x ,y > | 2 )
k
even
k odd .
As in the real case CMU], Green's formula and the harmonic property of the Hk's yield that spherical functions of different orders k and ] are L 2 (S n " t , du) -orthogonal. Thus for odd k and j , with m = Ck/2] and 1 = [j/23 and e€S n " 1 (C) 0=
J
Ok(<x,e>)
j
\<x
, e > |
2
cI>.(<X|e>) dw(x)
cp
m
(|<x,e>|2)
9i(l<x,e>|2)
dw(x)
Sn ~1(C) 1
= 2(n-l) J t2 9 m ( t 2 ) 9L(t2) o
t(l-t2)n"2dt
i
= ^
using
(v)
m = Ck/2],
J
and substituting ^(-y
1
)
is
(1-u)-2(l
-P,^)
+
u)du
u = 2t" - 1. This s h o w s that,
a multiple
of
(n
Pm ~
2>1)
(u),
for
hence
k odd with
and some
Konig: Complex Grothendieck constant in the n-dimensional case
189
M k €lR,
,
2 1 )
2
Y k i (x) Y k l (y) = F k ( < x , y > ) = M k < x , y > P m ( n " 2 > 1 ) (2| < x ,y > | 2 - 0
Z
= Mk Q k ( < x , y > ) . Take x = y to find M k = N k from the fact that P m ( n " 2 ' U ( l ) = 1 and the orthonormality of (Y k i ). The case of k even is similar; the missing factor < x , e > leads to P m ( n ~ 2 ' O ) instead of P m ( n " 2 ' ° .
•
3. The upper bound for KQ (n) We now prove that the upper bound for K ^ f n ) given in section 1 is valid. Proof of theorem 2. (i) Let A = (an) j "7= i sup { |
°e a (complex) matrix with §
aij
ti s, | | |t;| < 1, |sj| < 1 } < 1 .
(1)
Let N€lN. Averaging over the sphere S N ~ l (C) yields for x , y € S N ~ l ( C ) J S
s g n < x ,u > s g n < u , y > dd)N (u) = O ( < x , y > )
(2)
N - I
w h e r e
TC/2 J
O
2
cos
2
a. —
d a
V l - s sin cc
Given any elements x t , • • x m ,y 1} • • y m € S N Me) and replacing tits} by sgn < Xj, u > , sgn
aj; O ( < x i , y - i > ) | < 1
in (1)
(C) (3)
Haagerup CHa] shows this by Gaussian averaging. The spherical average in (2), however, is as easily deduced from real (2N)-dimensional averaging where the corresponding function 9 is cp(s) = — sin" relation 2TT:
sgn z = j
j
sgn Re (e~l* z) e1* dd
s CLP] - and the
190
Konig: Complex Grothendieck constant in the n-dimensional case
given
by
[Hal.
Using
this,
calculation
shows
that
the
left
side
of
(2)
equals
b
b
w h i c h is e v a l u a t e d t o be 0 ( < x , y > )
For
n€lN
We
will
such
consider define
the
Hilbert
nonlinear
a s in LHa3.
space
functions
H := l 2 ( L 2 ( S n ~ 1 ( C ) ) , d o ^ ).
12 - s u m
f,g:Sn~1(C)
>H
(depending
on
that I m f , I m g c S ( H ) = { h € H I llhll = 1 >
(O : ID
> ID is i n v e r t i b l e by [ H a ] if ID = {z € C I Izl < 1 } . )
By
= c]>-1(cn<x,y>cn)
Vx,
y € S
n-i
(4)
3o
( c )
H
•
(5)
l e t A b e a s in (1) and t a k e any e l e m e n t s Xj, • - x m , y j , • • y,^ € S n ~
Now
n)
(4),
the
elements
some N-dimensional
xi : = f ( X J ) , y j := g ( y j )
belong
to
the
unit
(C).
sphere
s u b s p a c e H N of H with N < 2 m . T h u s (3) and (5)
of can
be applied, yielding
c n 1^2
This implies
< x i > y j > | = |. . § a,, < D ( < x i , y j > H N ) | < 1 .
ajj
KG (n) ^ ^
(since
Xj.yj
may b e
taken
to
be
normalized
by
homogeneity).
(iii)It remains t o c o n s t r u c t f , g
w i t h ( 4 ) , ( 5 ) a n d t o e s t i m a t e c n . Let q k b e t h e
p o l y n o m i a l s g i v e n in s e c t i o n 2 . They w e r e o r t h o g o n a l with
respect
to
the
weight
function
w ( t ) = |t|
polynomials on n
(t-t~) ~
and
[-1,1] satisfy
l q k ( t ) | < 1 f o r | t | < 1. T a k e any 0 < c n <1. S i n c e c p " 1 e L 2 ( [ - I 1 l ] , w ( t ) d t ) ) vergent series
=
f k
«k(cn)
qk(t)
, t€C-l,13 .
(iv)
and
(6)
odd
Only t h e q k o f odd order o c c u r b e c a u s e 9 in
L2(w)-con-
expansion
(p~Hcnt)
see
w e have an
u s e now
that
(6)
* is an o d d
function.
We
shall
is an a b s o l u t e l y c o n v e r g e n t s e r i e s ,
i.e.
Konig: Complex Grothendieck constant in the n-dimensional case
Z
191
I a k ( c n ) |
k odd odd
for any x € S"" 1 and we can define f , g : S"" 1 f ( x ) : = (sgn a k ( c n ) / l a k ( c n ) | N k g(y):=( /lcck(cn) |Nk
> H = l 2 ( L 2 ( S n " 1 ) ) by
Q k ( <x , • > ) ) k
odd
Qk(
By the corollary to the addition theorem for spherical functions
=
H
Z
ak(cn) N
k —1 k odd
J
k
Qk(<x,z>)
Qk(
o n —1 **
oo
Z
ak(cn) Q k ( < x , y > )
=
(cn<x,y>)
.
T h e l a s t e q u a l i t y f o l l o w s f r o m ( 6 ) and t h e f a c t t h a t f o r z € ID ®~i(z)
= s g n z
, Q k ( z ) = sgn z q k ( z )
(k o d d ) .
T h u s ( 5 ) is verified. C o n c e r n i n g ( 4 ) , i . e . | | f ( x ) | | H = | | g ( x ) | | H = 1, w e g e t a condition on c n .
oo
Z k
Iak(cn) I Qk( < x , x > )
k=l odd
=
Z k
Iak(cn) | .
k=l odd
The absolute convergence of (6) ensures that, in fact, f(x),g(x) € H. To oo
satisfy ( 4 ) , we have to require
Z k
I a k ( c n ) 1 = 1,
i.e. we have to find
k =1 odd
0 < c n < 1 maximal with this property. Then K
(n) < -Q^ .
(iv)By Haagerup [Ha], for any 0 < c n < 1, there is an absolutely
convergent
expansion cp'^Cnt) =
Z j
3j cjl t j , t € C - 1 , l ]
(7)
odd
with 3i = 4/TT and 3j < 0 for j > 3 and odd. For j odd , write tj =
Z k = 1 k odd
Yjk ^k
(8)
192
Konig: Complex Grothendieck constant in the n-dimensional case
(The Yjk
are
non-zero only for odd k, since t' is an odd function.) >
We claim that Yjk
0 f° r
a
" °dd k < j : Since the q k 's are orthogonal
with respect to w, one has Yjk
=
^tJ »qk^w/^qk >^k^w
anc
* thus
l
sgn Yjk
=
s
gn (t ,qj < ) w = sgn I t q k (t) t(l-t")
" dt .
b Let 1:= Ck/23 , m*= [ j / 2 ] . Hence 1 < m and -^— = m + 1. The definition of q k yields after substitution of s = 2 1 2 - 1 i
s g n Yjk
P j ( n " 2 f l > (s)
= sgn J
(l-s)n"2 d + s)
m + 1
ds .
-1
By Rodriguez' formula CSz] 1
n1
L \ I /
with c n j > 0. Hence 1
s g n Yjk
=
s
g n (~1)
J
D [(1-s)
+n
" (1 + s )
+
] (l + s )
m
d s
-1 l
= sgn J
d-s)l
+n
" 2 d + s)1+ 1 Dl[(l + s ) m ] d s
after m - f o l d integration. Since I < m, the integrand is strictly positiv and thus Yjk
>
^ f ° r a^
For t = l ,
oc
ld k < j .
( 8 )yields 1 =
S
Yjk =
2
I Tji< I- I n s e r t i n g
( 8 ) into ( 7 )
odd
thus gives an absolutely convergent expansion of cp~1(cnt) in terms of the
j odd
k
odd
cn) k
odd
j odd
k
qk(t)
odd
CO
which is (6) with a.,(c n ) =
2 j
3s Ysi^ cj..
Since YIL- > 0 and
B-. < 0 for
odd
j > 3 odd, we conclude that a k (c n ) < 0 for all k > 3 odd and 0 < c n < 1. On the other hand a t (c n ) > 0 since for t = 1
Konig: Complex Grothendieck constant in the n-dimensional case
193
odd
k
oo
with 9
1
(cn) > 0 on (0,13. In particular, oc^l) = 1 - £
a k ( l ) > 1. The sign-
k —3
character of (<xk(cn)) is thus the same as the one of ((3k). The condition on oo
Z I <*k(cn) I = * > ' s equivalent to
cn , k
odd
1 = 2at(cn) -
a k ( c n ) = 2 a t ( c n ) - 9~ 1 (c n ) .
^
(9)
odd
k
(v) To evaluate ocj(c n ), put Aj(n):=Y:j (which depends on n, t o o ) . Note that q t ( t ) = t. For odd j and I = Cj/23, calculation shows 1
l
Aj(n) = ( t j , t ) w / ( t , t ) w = J t j + 2 d - t 2 ) n " 2 d t / J t 3 d - t 2 ) n " 2 dt o o =
A (n) = l
"(nil)!* '
l
For this value of Aj(n) , a t (c n ) =
f j
'
Aj(n) (3j c£. Since 9 : CO,t]
> [0,13
odd
is bijective, we can put s n =
2at(
(10)
odd
We claim that (10) has exactly one solution s n € ( 0 , l ] . To prove this, let f ( s ) : = 2 a t ( < p ( s ) ) - s.
Note
that
f(0) = 0
and f ( l ) = 2a 1 (1) - 1 > 1 since
a^U) > 1 as seen in ( i v ) . Thus it suffices to prove that f is strictly increasing on (0,1 ] . We show that f' > 0 on [0,1]. First of all, cp'(s) > 7t/4 for s € [0,1] as easily seen from the series expansion of the function 9 , cf. [Ha]. Thus, using
pj = 4/7t , (3j < 0 for j > 3 , A , ( n ) = 1 and
f ' ( s ) = 2
odd
j
*#
A j ( n ) Pi j 9 ( s ) j " 1 - 1
• S
(^ " ,|3 j
A
» ( n ) • IPJO - * •
<">
odd
Now Aj(n + 1) = n ^ ^ c j / 2 3
A
i(n) -
A
i(n)
and
thus
f o r a11 n
-
2
194
Konig: Complex Grothendieck constant in the n-dimensional case
Aj(n) We have
j < Aj(2)
£
j = - 5 7 ^ 1
=^
< 4
IfyI = -| - 1 since
j odd
J
j=1 j odd
j=3 j odd
J
Using | p 3 | = -^ , we get 1
A,(n) j 13j I < 2 | p 3 | + 4 J
j- 3 j odd
j
= 4
S
3
j- 5 odd
|Pj|
I £3j 1 - 2 | p 3 | =
j odd
This and (11) yield f'(s) Hence
(10)
> 2TU + ^ has
- 7 > 0.1 > 0 .
exactly
one
solution
sn€(0,l). oo
Aj(n+1) < Aj(n), we have with
f n (s)
:=
Z
Since
At(n) = 1
and
Aj(n) fy cp(s)J - s
that
J
j odd
f n ( s ) < fn
+t(s)
and t h u s
1 = fn
+ 1 ( s n + 1)
= f n ( s n ) < fn
+1(sn).
an increasing function, ( s n ) n > 2 is decreasing and ( c (*s
)
Since
fn
is
+1
) n > 2 is increasing.
This e n d s t h e proof of t h e o r e m 2. We end
this
section
with
• a few
comments
on
the
conjecture
section 1. As mentioned t h e r e , Haagerup CHa] conjectures t h a t K G
in
< l/|
1
Now |cp(i)| = c iff 9 ~ ( i c ) = i iff
f
k=1 k odd
( - l ) C k / 2 : i (3k c k = 1 .
(12)
This would replace his proven bound KG < 1/(p(s o ) where cp(so) = c is determined from f
k=l k odd
lPkl c k = 1 .
(13)
Konig: Complex Grothendieck constant in the n-dimensional case
195
Since p s = 0, the first term where the series (12) and (13) differ is | p 9 | c 9 which is already fairly small; thus there would be only a small improvement of the KG -bound. The analogues of the coefficients (3k c k in the n-dimensional case oo
are the coefficients a k ( c ) , with condition (13) replaced by
Z k
| a k ( c ) | = 1.
odd
Thus (12) ought to be replaced by f (-l) Ck/23 <xk(c) = 1 , k=l k odd
(14)
and then KG (n) < 1/
Pj Tjk
§
cJ
.
k
odd
j= k j odd
where the coefficients Tjk = T]k^ n ^ °f
tne
expansion of t J in terms of the
polynomials q k can, be calculated explicitely, / \ - / j.o \ Tik —; KV n ) = ( n + z m )
'J
1? —
(l-m)!m!
(1 + 1 ) ! —
(m + n - l ) ? —
(1 + m + n ) !
(m + n-2) ! ——; —-
(tn+1) ! ( n - 2 ) !
with m = Ck/2] , 1 = Cj/2]. Equation (14) then reads 1 =
where
(-l)Ck/23 ak(c) =
g
B:(n):=
f
J
k
k
= l odd
(-1)Ck/2D
Tjk(n) iR
I
B,(n) P, c j
= (-l)Cj/23 ^ l
^'V,"'^~]\,
(n + 1 ) ( n +3 ) - ( n
+ j-2)
If this conjecture were true, it would be in complete formal analogy with the results of Krivii Krivine CKr] in the real case of KG (n). The equations for n = 3,5 ,7 would be P,c=1
, P, c + | p 3 | / 3 c 3 = 1 , p, c + IP3I/2 c 3 = 1
noting that p s = 0. This would give bounds K G C (3) < 4/TC , jJk - 3//9T7T-3)
The difference to the proven bounds is about 10
for small values of n.
K
196
Konig: Complex Giothendieck constant in the n-dimensional case
4. Some remarks on the lower bounds for K& (n)
A slight variation of A. M. Davie's very clever method for the lower bound KG
> 1.338 CDa] yields lower bounds for K G ( n ) . Unfortunately, Davie's arguments
are still unpublished. I would like t o thank G. Jameson for sending me s o m e notes on this lower bound. A few remarks on the lower bounds for K G (n) might be in order, nevertheless. For 1 < p < oo, let Lpn = L p ( S n " 1 ,do) n ) and P n denote the orthogonal
projection n
E n = Span
{<•, x>|x€lK }
(in L2n) onto
of (restrictions
integral operator with kernel n < x , y > It
is
easily
checked
that
on S n
the n-dimensional
of) linear functions.
, it may act between any L " - s p a c e s . > L2n II < / n .
|| P n : L"
subspace P n is an
Let
0 < p < 1 and
T n = T n ( p ) = P n - p Id. Denoting the nuclear and 1-summing norms, respectively, by v and TCJ, respectively, w e have / n < v ( P n : L<£ n = tr(Pn)
< ||P n : L2n
> h£) since > L2n)
> LcSH v ( P n : L<£
and thus, using ( l - p ) P n = P n T n , ( 1 - p ) /H < v ( ( l - p ) P n : L ^
> Ltn || 7c,(P n : Ltn
< || T n : L ^
< K G ( n ) 7H | | T n : L ^ Thus with M p n := l|T n (p) : L ^
> L2n)
>L?\\
>L 1 n || , K G (n) >
sup O
Taking
p = 0,
one
gets
for
>L 2 n )
>L£) = ^ ( ( 1 - p ) Pn : L^
n
=• oo
the
^
[V1
. p
classical
lower
bounds
KGR >K/2 , KGC > 4/TT. Explicitely Mpn
=
P
sup
I
f
'snJ-l
|f|,|g|
f n < x , y > da)(x) d w ( y ) - p J
sn
f f (x) g ( x ) d o ( x ) I. J
-\
sn
-\
&
'
Davie ingeneously estimates this: the projection P n leads to a linear function h e (x) = Vn < x , e > with e € S n ~ 1 ( C ) such that in the complex case Mj1 <
sup {(
where
^(x) = (
,
t
v
r | h
* (x) |he(x) I d x ) 2
f
e
( x ) | t
/ [ i
| h
e
( x ) |
< (i i
+
p - 2p
M
|* ( x ) I2 d x }
f
,
|h ( x ) | > (i J " A c t u a l I y . equality holds.
Konig: Complex Grothendieck constant in the n-dimensional case
197
This is clearly independent of e € S n " 1 ( C ) . The integrals can be calculated by using the formula in (v) of section 2. Rescaling X = (i/-/n and differentiating with respect to X shows that the maximum is attained for 0 < X < 1 determined by p = p(X) = 1 - ( 1 - ^ 2 > " ~ 1
(Davie takes n
- -H^t X 2 ( I - * 2 ) " " 1 • n ( n - l ) X j t 2 ( l - t 2 ) " - 2 d t . (15)
> oo, working with error integrals.) The last equation is
uniquely solvable for 0 < X < 1 provided that 0 < p < 1/2; only this range of p's turns out to be of interest. Inserting this value of p = p(X), one finds l
Mp(nx) = P(X) ( l - 2 ( l - X 2 ) n ' 1 s u p -LZL£- =
Now KGC(n) >
sup
O<<1/2 O
expression —kL_— M
M^1 M^1
c a n De
+
2
ilazll J s u p1~
sup
CXX<1 CXX<1
M M(
p i
nn
t
(1 - t 2 ) " " 2 dt ) .
)
(16)
• Using formulas (15),(16), t h e
X ) p(X)
calculated for any 0 < X< 1. Numerical evaluation
p(X)
yields t h e b o u n d s given in t h e t a b l e in section 1. The optimal value of p = p(X) does n o t depend very s t r o n g l y on n € IN and is close t o p ^ . 23 .
References [Da]
A. M. Davie; Unpublished notes.
CDGS]
P. Delsarte, J. M. Goethals, J.J.Seidel; Bounds for systems of lines and Jacobi polynomials, Philips Research Reports 30(1975), 91-105.
[Ha3
U. Haagerup; A new upper bound for the complex constant, Israel J. Math. 60 (1987), 1999-224.
[Ko]
T. H. Koornwinder; The addition formula for Jacobi polynomials and spherical harmonics, SIAM J. Appl. Math. 2 (1973), 236-246.
[Kr]
J. L. Krivine; Constantes de Grothendieck et fonctions de type positif sur les spheres, Advances in Math. 31(1979), 16-30.
[LP]
J. Lindenstrauss, A. Pelczynski; Absolutely summing operators in £ p - s p a c e s and their applications, Stud. Math. 29(1968), 275-326.
[LT]
J. Lindenstrauss, 1979.
[Mii]
C. MUller, Spherical Harmonics, Lect. Notes in Math. 17, Springer, 1966.
L. Tzafriri;
Classical
Banach Spaces
Grothendieck
II, Springer,
198
Konig: Complex Grothendieck constant in the n-dimensional case
[Pi]
G. Pisier; Grothendieck's theorem for non-commutative C*-algebras with an appendix on Grothendieck's constant, J. Funct. Anal. 29 (1978), 379-415.
[Sz]
G. Szego; Orthogonal Polynomials, AMS 1959.
CTJ]
N. Tomczak-Jaegermann; Banach-Mazur sional operator ideals, Pitman, 1989.
Mathematisches Seminar Universitat Kiel Olshausenstr. 40 2300 KIEL W-Germany
distances
and
finite-dimen-
Pathological properties and dichotomies for random quotients of finite-dimensional Banach spaces P I O T R MANKIEWICZ
NICOLE TOMCZAK-JAEGERMANN
Institute of Mathematics Polish Academy of Sciences Warsaw, POLAND
Department of Mathematics University of Alberta Edmonton, Alberta, CANADA
1
l
Introduction and preliminaries
In this note we study Banach-Mazur distances and basic constants of proportional dimensional quotients of an arbitrary n-dimensional Banach space X. The main result establishes a dichotomous behaviour of these invariants for families of random quotients. It is shown that every space X has a quotient X\ such that either X\ is Euclidean or otherwise basis constants of random quotients of X\ are large and, for a random pair of quotients, the Banach-Mazur distances are also large. This provides a strengthening of some results from [M-T]. Our notation follows [M-T]. In particular let (X, || • ||x) be an n-dimensional Banach space. The unit ball of X is denoted by Bx- Let | • | 2 be a Euclidean norm on X and let (•, •) be the associated inner product. Identify (Xy | • | 2 ) with JZn under the natural Euclidean norm. Let 1 < I < n. Denote by Gnj the Grassmann manifold of all ldimensional subspaces of Rn with the Haar measure /in>/. Fix 0 < 7 < 1. For a family of properties M = {Mn} of 7n-dimensional spaces we say that M. is satisfied for a random subspace, if there exists 0 < 6 < 1 such that hnnn{E e Gnan I E has Mn} > 1 - 6n, 1 Supported in part by National Sciences and Engineering Research Council Grant A8854.
200
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
for every n = 1,2, We say that M is satisfied for a random quotient if the set of E G Gfn,(i«7)n such that X/E satisfies M has measure larger than or equal to 1 — 6n. Similarly we define a random pair of subspaces and a random pair of quotients. Let P be an orthogonal projection on (X, | • I2). The quotient space Y = X/kerP can be identified with the space P(X) under the norm ||J,||PW
= inffllj, + *\\x I z € kerP}
for y G P{X).
Notice that with this identification Y has a Euclidean norm inherited from -X", namely | • I2. Another representation of quotient spaces will be described in Section 2. Define the norm || • ||* on X by \\z\\*=suv{\{x,z)\\xeBx}
for z eX.
(1.1)
Then the space (X, || • ||») is clearly isometric to the dual space X*. Finally, for detailed information on random subspaces and measure concentration phenomena we refer the reader to [M-S]; on various ideal norms and Banach-Mazur distances—to [T]; and on recent geometric and related random methods in the local theory of Banach spaces—to [P]. The second named author would like to thank Bill Johnson for valuable conversations by E-mail concerning Lemma 3.4.
2
Estimates for spaces in special positions
In this section we discuss the space Rn with an arbitrary norm || • || and we denote this space by X. The unit ball of X is denoted by Bx or by Bni to emphasize the dimension. Recall that the norm from l\ is denoted by || • U2, the Euclidean unit ball is denoted by B\ and { e i , . . . , e n } denotes the standard unit vector basis in Mn. Let 1 < k < n. By Rk C Rn we denote span{e x ,..., e*}. By L(Rk) we denote the space of all linear operators on R k. For every (n — fc)-dimensional subspace E of Rn by PE we denote the orthogonal projection on EL. For every (n — A:)-dimensional subspace E of Rn, which does not intersect Rk, by QE : Rn —> Rk we denote the projection onto Rk with the kernel E. By QE(X) we denote Rk with the norm for which the unit ball is QE(Bn). That is, \\y\\QBlx) = in£{\\y + z\\ \ z G £ } , for y G Rk. In
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies 201 particular, in this section we will identify the quotient space X/E
(where
E n Rk = {0}) with QE(X). For 1 < k < m < n set Vm,k = Vm,k(X) =
(2.1)
sup EeGn,n-m
inf
(volk(F fl PE(Bn))/volk(F
D PE(Bl)j)V*
FCEJ- ,dimF=* V
.
'
We write Vk for Vkyk. The quantity T^* is closely related to the notion of the volume ratio numbers of operators, which will be of importance later on. Let us recall ([M-P], [P-T], [P]) that for an operator u : X - • l2n and 1 < j < n, the j - t h volume ratio number is defined by = sup{(volk(PE(Bx))/yolkPE(B2n))1/k
\EcMn,codim^
< j}. (2.2) In particular, Vk(X) = vr n _k(i), where i : X —• \2n is the identity operator. Random subspaces of JRn enjoy a number of good geometric properties which axe fundamental in this investigation. These properties were discussed in details in [M-T]; here we only list the ones which are directly used in our arguments. VTJ(U)
Fact 2.1 Let 0 < 7 < a < 1. There exist ct — ci(a,j), c2 = c2(a) and e = £(7) > 0 such that a random subspace E G G>i,(i-cr)n ^as ^ e property: (A) The operator QE [E1 : EL —> Ran has at most jn s-numbers larger than c\. (B) ||Q.Eet-||2 < c2) for i = an + l,cm + 2 , . . . ,n. (C) dist (Q^CnSpanlQ^ejt | (1 — j)n < k < i}) > e, for (l—j)n
< i < n.
Let (fi, P) be a probability space. We shall say that a Gaussian random variable in a A;-dimensional subspace E of Rn has distribution N(0,1, E) if its density (with respect to the Ar-dimensional Lebesgue measure on E) is equal to f(T\
W2z>-*llxll2/2
—(
fnr
T £= F
2?r
In the sequel we shall need the following well known basic properties of such random variables (c/. e.g. [Sz]).
202
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
Fact 2.2 Let E be a k-dimensinal subspace of Mn and let g(u) be a Gaussian random variable with distribution AT(0,1,12). Then (D) For every orthogonal projection P in E onto an m-dimensinal subspace F, \/k/nP(g(u)) is a Gaussian random variable with distribution i\T(0,l,F). (E) For every pair F\, F2 of orthogonal subspaces of E, the random variables PF±(g(u)) and PF±(g(u>)) are independent. (F) There is a universal constant a such that P ({« € ft 11/2 < \\g(u>)\\2 < 2}) > 1 - c - B , for every Gaussian random variable g with distribution N(0, l,J2 n ). The following two lemmas are well known to specialists. Lemma 2.3 Let B be a centrally symmetric convex body in Mn and let g be a Gaussian random variable with distribution N(0,l,Rn). For every operator T G L(Mn) we have n-i ({x e S71"1 I Tx e B}) < 2P ({a; 6 fi | Tg(u>) E 2B}). Here /i5n-i denotes the normalized Haar measure on the unit sphere 5 n - 1 in Rn. Proof Observe that the measure v defined for a Borel subset C C 5 n - 1 by u(C) = jP ({u, € a I |b(u;)|| 2 < 2 and
g(U)/\\g(u,)\\2 6 C}) ,
where A = P({w G ft | |
and
Tg(u) e \\g(u,)\\3B}) •
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies 203
Lemma 2.4 Let E be a k-dimensional subspace of Rn and g be a Gaussian random variable with distribution JNT(0,1, E). Then for every random variable y — y(u>) independent of g and taking values in Rn, and for every centrally symmetric convex body B C Mn we have P ({u e ft | g(u) eB + y(uj)}) < a* volk(E H B)/vo\kB2k. where a\ > 1 is a universal constant. Proof It is enough to prove the lemma with y being a constant variable; the general case is then obtained by an appropriete integration. Clearly, P ({a; G ft | g(u>) G B + y}) = P ({a; G ft | g{u>) eEn(B
+ y)}).
(2.3)
By Lemma 6.6 in [Sz], the latter probability is smaller than or equal to
where a\ > 1 is a universal constant. Obviously, \o\k(E fl (B + y)) = voljt((-E7 — y) fl B). It is a well-known consequence of the classical Brunn inequality (c/. e.g. [H]) that the latter volume is the largest if y = 0. That is, the latter quantity in (2.3) is smaller than or equal to vo\k(EnB)/vo\kBk, • completing the proof. The next proposition gives an upper estimate for probability of certain sets in terms of the invariant Vmik{X). Proposition 2.5 Let E be a fin-dimensional subspace of Mn with 0 < /3 < 1 and let g be a Gaussian random variable with distribution iV(0,1,-E). Let B be a centrally symmetric convex body in Rn. Let S : E —> S(E) C Rn be an operator with all s-numbers greater than or equal to a > 0. Let 0 < 6 < 1. For every k with 6/3n < k < f3n we have
P ({« Gft| Sg{u) G PS(EAB)})
< {ca^V^B))"
,
where c = c(6). Proof Fix k satisfying Sfin < k < ftn. By the definition of Vpnjk there exist a subspace F C S(E) with d i m F = k such that (vol,(F H Ps(E)±(B))/volkBl)1/k
< VM(B).
(2.4)
204
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
Set Fi = S"1 (F). Let F2 be the orthogonal complement of i*\ in E and let Pi and P2 be the orthogonal projections in E onto i*\ and F2 respectively. Write g = Pxg + P2g. Note that, by Fact 2.2 (E), the random variables Pig and P2 are independent. Moreover, by Fact 2.2 (D), Jfin/kPig is a Gaussian random variable with distribution N(0,1,-Fi). Therefore, by Lemma 2.4 we have
(2.5)
<
4^\Jfj-s-\FnP8(E)±(B))\
where ai is a universal constant. On the other hand, by (2.4), easy computation yields
Ps{E)x(B))/vo\kBl (2.6) which combined with (2.5) concludes the proof.
•
We come now to a crucial point of this section which is a refinement of Lemma 3.3 in [M-T]. To state this very technical estimate some more notation from [M-T] is required. Fix 0 < a < 1. Set T= {E e G n , (1 _ a)n | E satisfies (A), (B), (C)}. Let 0 < /? < a, and 7 < min(o:, 1 — a), and rj > 0 satisfy a — (7/ + 2j) > /?. Denote span{ei,... ,e 7n _x} by ]Rin~l and let Q denote the group of all
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies 205
W G O{n) such that
W\R^n-1 = Id and W\{RmY = Id.
(2.7)
By hg denote the normalized Haar measure on Q, Lemma 2.6 Let 0 < 6 < 1 and let A > 0. Let X be an n-dimensional Banach space. Let EQ G G n) (i- a ) n satisfy (A) and let Ban = QE0Bn. Fix an arbitrary Fo G 7 and arbitrary U G O(n) such that U(F0) G T. Then we have hg{W e Q\TWQu{Fo)en G ABan)} < (ce~lAV0n,kf, for every k with 8fin < k < /3n and for every T G L(MQn) which has exactly rjn s-numbers smaller than 1; where e satisfies condition (C) and c = c(a,P,6). Proof By condition (A), E£ = EX®E\ with Ex _L E'u QEo(Ei) -L QEO(E'I) and dim-Bi < 771, and such that QE0 \ Ei (resp. QEQ \ E\) has all 6-numbers larger than cx (resp. smaller than Ci). Set F± = QEO(EI) C Ran. Fix an operator T G L(Man) which has exactly r\n s-numbers smaller than 1. Write Man = E2®E'2 with E2 ± E\, T(E2) _L T(E'2) and dimE 2 = 7yn, and such that T | E2 (resp. T \ E'2) has all 5-numbers smaller than 1 (resp. larger than or equal to 1). Set F2 = T(E2) C Ran. Set F3 = T(R^n~l), so that dimF 3 <
for j = 1,2,3.
The existence of P like this follows from the condition a — (7/ + 2j) > fi. Set G APBan). A = {W G Q I PTWQumen Clearly, if A denotes the set in the conclusion of the lemma, then A C A'.
(2.8)
Denote by H the orthogonal complement of Rin~l in Ran and by Pi— the orthogonal projection on H. Since kerPi C kerPT then PT = PTPX. Also, WPX =PXW for every W G Q. Set Z = Ql/(F 0 ) e n
206
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
and Z0 = P1(z)/\\P1(z)\\2. We have PTWQu(Fo)en = PTW{zo)\\Pi{z)\\2. Recall that U(F0) 6 T and in particular it satisfies condition (C). Let e > 0 be a constant as in (C). Then ||Pi(*)|| a > e. Set A" = {W € Q | PTW(zo) € e^APBan}.
(2.9)
A' C A".
(2.10)
Thus Since £/ can be identified with the orthogonal group acting on H, the measure of A" is equal to the measure of a suitable subset of the sphere S# in H. We have ho(A") = nsH ({x e SH I PTx G e-lAPBan})
.
(2.11)
Denote by A the subset of SH which appears on the right hand side of (2.11). Let P2 be the orthogonel projection in H with kerP 2 = ker (PT\H). By Lemma 2.3 we have <
2P({ojeSl\PTg'e2e-lAPBan})
= 2P ({a; e ft | PTP2g' £ 2e'1APBan}) ,
(2.12)
where g' is a Gaussian random variable with distribution N(0,1,H). Set E = P2(H) and g = ^ ( a - 7 + 1)//?P20'. Clearly, dimJ5 = fin and, by Fact 2.2 (H), g is a Gaussian random variable with distribution N(0,1,E). By (2.12), we infer that 2P ({« 6 ft I PT<7 G c T M P B a n } ) ,
(2.13)
where c' = 2y/a — 7 + 1. Consider now the operator Q = QEo\E£ : E£ -> RQn. Recall that PEo denotes the orthogonal projection onto EQ. Therefore QE0 = QPE0 and P B a n = PQE0BTI = PQPE0Bn. Clearly, Q is an isomorphism. There is a projection P 3 in Man such that kerP 3 = kerP and that P' = Q~lP^Q is an orthogonal projection in EQ of rank fin. In fact, P 3 is the projection onto
Mankiewicz & Tomczak-JSgermann: Pathological properties and dichotomies 207
)-1. In particular, P3(Ran)
C Q{E\).
Clearly, P 3 = P3P, so
that P = Q^PZPQPE,
(2.14)
is an orthogonal projection of rank /?n with ker P D Eo. Set 5 = Q^PzT\E:E-+
S(E) C Eft. Then
C{ueSl\Sge c'e^APBn}.
(2.15)
Observe that PS = 5, so since rank S = /?n, P is the orthogonal projection in Rn onto S(E). We will show that all s-numbers of S are larger than or equal to cj"1, where c\ is the constant from condition (A). By Proposition 2.5 this will imply that the measure of the latter set in (2.15) is smaller than or equal to {ca~xVpnrk(B))k. We will then conclude the proof combining this estimate with (2.13) and (2.15). To estimate s-numbers of 5, let R = T \ E. Since E C E[, all s-numbers of R are larger than or equal to 1. We have PT \ E = R(R"1PR) = RP", where P" = R~XPR is a projection. Clearly, kerP" = kerP 2 -L£, hence P" \ E has all s-numbers larger than or equal to 1. Similarly, since P(Ran)± kerP3, then all s-numbers of P3\P(Ean) also are larger than or equal to 1. Finally, since P(Ran) C QEO(E[) then all s-numbers of Q~1\P(Ran) are larger than or equal to c\. Now recall that for any operator U : Rm —> J? m , all s-numbers of U are larger than or equal to c > 0 if and only if the inequality \Ux\ > c\x\ is satisfied for every x £ Rm. (Here | • | denotes the Euclidean norm on Rm.) This implies that all s-numbers of the operator S = Q~lPzRP" are larger than or equal to cj"1, completing the proof. • Lemma 2.6 is the only modification required to get strengthenings of the main technical results of [M-T], Sections 3 and 6. Similarly as in [M-T], we are able to obtain estimates for distances and basis constant for random quotients of n-dimensional spaces in a special position. In fact we shall state the result for spaces which additionally admit a good control of volume ratio, leaving a general case to an interested reader.
208
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
Let us recall that a space X = (-E n ,|| • ||) is said to be in a special position if a\\x\\2 < \\x\\ < M i for xeX, (2.16) for some constant a > 0. (Here || • ||i and || • ||2 denote the norms in l\ and l\ respectively.) Recall that the volume ratio of X, vr (X), is defined by vr (X) = (vol Bx/vol £)1/n ,
(2.17)
where £ C Bx is the ellipsoid of maximal volume contained in the unit ball BxoiX. The main result of this section is a refinement of Theorems 3.5 and 6.5 from [M-T], Its proof repeates the arguments from Proposition 3.2, Theorems 3.5, 3.7, Proposition 6.1, Theorem 6.5 and Proposition 4.2 in [M-T], and combines them with Lemma 2.6 above. Theorem 2.7 Let X be an n-dimensional Banach space satisfying (2.16) with the volume ratio vr(X) < A, for some constant A > 1. Let 0 < a < I, let 0 < /? < a/2 and let k = 6f3n, for some 0 < 6 < 1. For a random pair (£7i, -E/2) 0/(1 ~ ot)n-dimensional subspaces of Rn one has d(QEl(X),QE2(X))
> cA-c'V£k.
(2.18)
Furthermore, set m = min(3a/2 — /?, l)n and let Xm = Pm(X), where Pm : Rn —> JRm is the orthogonal projection onto Rm. Then for a random (m — an)-dimensional subspace E of Rm the following basis constant estimate holds bc(QE(Xm)) > cA-<W£k. (2.19) In general one has c = c(a,fi,6) > 0 and c1 = c'(a,^,<5) > 1. If for some 0 < K < 1 one has a < K, /3 < /ca/2 and 6 < K, then c = c(/c) and d = d(n).
3
Dichotomies for random quotients
Now we are ready to study proportional dimensional quotients of an arbitrary n-dimensional Banach space X. Actually, we will be mainly interested
Mankiewicz & Tomezak-Jagermann: Pathological properties and dichotomies 209 in properties shared by random quotients with respect to some Euclidean norm on X. The main result of this note says that every space X has a quotient X\ which satisfies the dichotomy: either X\ is Euclidean or potherwise basis constants of random quotients of X\ are large and, for a random pair of quotients, the distances are also large. The notion of random quotient obviously depends on the Euclidean norm on X , and the result will be established for a large class of such norms. Denote a Euclidean norm on X by | • | 2 and let [•, •] be the associated inner product. Assume that | x | 2 < ||s||
forxGl.
(3.1)
Moreover, in the present context it is natural to assume that the Euclidean unit sphere and the unit ball of X are close to each other in some directions, and that these directions are equally distributed in the space. Perhaps the weakest assumption of this type is the following. Fix absolute constants 0 < £, #i < 1. Assume that there exist vectors y i , . . . , y/, with / = 8n, which are orthonormal in | • | 2 and such that | | y i | U = S U p { | [ a ; , y i ] | | ||.r|| < 1} > £x
for j = 1 , . . . , / .
(3.2)
Note that if | • | 2 is the norm determined by the ellipsoid of minimal volume containing the unit ball of X , then for every /era-dimensional subspace E of X there is y 6 E with |y| 2 = 1 satisfying (3.2) with a\ = yJH. Now our result states. Theorem 3,1 There is a constant 0 < a0 < 1/2 such that the following holds, for every 0 < a < ao- Let X be an n-dimensional Banach space, and let | • | 2 be a Euclidean norm on X satisfying (S.I) and (3.2). There is an integer lottfi < m < n and an m-dimensional quotient X\ of X such that for every K > 10 exactly one of the following two conditions holds. (i) A random pair (Fi,F2) of an-dimensional quotients of X\ satisfies d(F1,F2)>K2;
(3.3)
and a random an-dimensional quotient F of X\ satisfies bc(F)>K.
(3.4)
(ii) There exist constants c\,C2 > 0 such that c\c2 < K and that (l/ci)\x\2
< \\x\\Xl < c2\x\2
for
x e Xx.
(3.5)
210
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
The proof of the theorem is based on Theorem 2.7 and powerful methods from the local theory of Banach spaces. It requires some additional results. The first one is analogous to [M-T] Proposition 4.3. It ensures that the assumptions of Theorem 2.7 are satisfied for some quotient of X. Proposition 3.2 There exist constants 0 < a0 < 1 and 1 < aya!,a" with the following property. Let X be an n-dimensional Banach space, and let | • I2 be a Euclidean norm on X satisfying (S.I) and (S.2). There exists a quotient of X, say Y = P(X), for some orthogonal projection P, with dim Y = k > aon, such that vv(Y)
(3.6)
Furthermore, there exist a Euclidean norm || • 112 basis { e i , . . . , e*} in (F, || • ||2) , such that
^
on
Y'j
an
an
d
orthonormal
V = I>*ey,
(3.7)
for
(3.8)
and (l/«")|y|a < IMI2 < a"|y|2
y € Y.
Remark If | • | 2 is the norm determined by the ellipsoid of minimal volume containing the unit ball of X, then the above proposition holds for every 1 < k < n. Constants a, a' and a" depend then on 6 = k/n. Proof For every j = 1 , . . . , / , pick Xj G X such that ||XJ|| = 1 and that |[#j,yj]| = sup||x||<1 |[z,yj]|. Using Bourgain-Tzafriri's method ([B-T] Theorem 7.2, cf also [B-S]) one can choose a subset a C { 1 , . . . , / } , with \a\ > 2- x % 2 / such that /
E N ^ IIE^II > l E ' ^ b >(*i/4) El^l t'€
t'€a
i£
\i£a
\i/2
2
>
(3-9)
/
for any sequence (t t ) of scalars. Denote by Yi the space span{xt}t6<7 with the quotient norm given by the orthogonal projection from X onto Yi. Clearly, the norm || • ||yj also satisfies (3.9). It was proved by V. Milman, as a consequence of his inverse Brunn-Minkowski inequality [M.I], that Y\ has a quotient Y2 of dimension dimF 2 = dimYi/2, say, such that vr(F 2 ) ^ 5-
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies 211
Represent Y2 as an image of Y\ (and also of X), by some orthogonal projection, say R. Now proceed as in Proposition 4.3 of [M-T]. Let 5 be an orthogonal projection in Y2 with rank S > a2 dim Y2 such that
where 0 < a2 < 1 < a are absolute constants. Required vectors e t 's can be chosen among the SRxSs. The space Y is the span of the e t 's under the quotient norm, and the new Euclidean norm || • ||2 on Y is set to be II £,*.e,|| 2 = (£,• M 2 ) 1 / 2 . Then (3.7) and (3.8) follow from Lemmas D and C in [B-S] by the same argument as in [M-T]. Q We also need some estimates for Gelfand numbers of operators from a Banach space X to a Hilbert space, in terms of volume ratio numbers of u. Let us recall that for u : X —> Y the m-th Gelfand number is defined by cm(u) = inf{||w | JS|| | E C X, codimE < m}. Recall that the volume ratio numbers were defined in (2.2). The following result was observed by Pajor and Tomczak in [P-T]. It is close in spirit to the volume ratio result of Szarek. Proposition 3.3 Let X be an n-dimensional Banach space and let u : X —* /£. For every m = 1 , . . . , n one has c2m(u) < a2(n/m) log(l + n/m)vr m (ti),
(3.10)
where a2 > 1 is a universal constant. Let us provide an outline of the proof of (3.10). Proof Define the Euclidean norm on X by |x|2 = \\ux\\ for x G X. It is clearly enough to prove (3.10) for the formal identity operator i : X —• (X, | • | 2 ). Fix m. By Milman's theorem on quotients of subspaces [M.2] (cf. also [M.I], [P]), there is an m/2-codimensional subspace E C X and an (n — m)-dimensional quotient F of E such that d(F, /£_m) < D} where D = a(n/m) log(l + n/m), for some universal constant a. Represent F as P(E), for some orthogonal projection P , and denote its quotient norm by || • ||/>(#)• The required inequality is easy to establish
212
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
for operators acting between Hilbert spaces. In particular (cf. e.g. [P-T]), there is a subspace H C P(E) with dim H = n — 2m such that
< Psup J%ff%f g) MP(E) forxeH,
(3.11)
TI—in/2/
with the supremum taken over all orthogonal projections Q : P(2£) — of rank n — 2m. Consider now F = P(-B) as a subspace of 25, that is, F with the norm || • ||, inherited from E. Clearly, ||z||p(£) < ||z||, for x G F. Furtheremore, if R is the orthogonal projection on 22, then VO\QP(BE)
< volQPR(Bx).
Substituting these estimates into (3.11) and observing that (if, || • ||) is a subspace of X and QPR is an orthogonal projection in X, we conclude the • proof of (3.10) by the definition of volume ratio numbers. Finally, we also require the following simple lemma, known to specialists. Lemma 3.4 Let X be a k-dimensional Banach space such that d(-Y, /£) ^ D, for some D > 10. There exists a (0.01k)-dimensional subspace E C X such thatd(E,ll01k) < (1/2)2). Proof Let || • ||2 be a Euclidean norm on X satisfying M|2<||*||
for* € X .
(3.12)
Let Mr denotes the median of || • || on the unit shere {x G X \ ||rr||2 = 1. The classical proof of Dvoretzky's theorem (cf. e.g [M.3], [F-L-M] or [M-S] Theorem 4.2) shows that for a fixed e > 0, whenever m < (in^^l^l^Tr^)
+ e2(k - 3)/2],
(3.13)
there exists an m-dimensional subspace E C X such that for x G E one has [(2/3)Mr - (4/3)Z>e]||*||2 < \\x\\ < (4/3)(M r + De)\\x\\2.
(3.14)
Now, set e.g. e = 3/40, so that m = O.Olfc satisfies (3.13). If Mr > 0.3D then (3.14) implies that for some m-dimensional subspace E C X one has
Mankiewicz & Tomezak-Ja'germann: Pathological properties and dichotomies 213 where K = D/Mr. Otherwise, combining the upper estimate in (3.14) with the lower estimate in (3.12) we get that some ra-dimensional subspace E C X satisfies
completing the proof.
•
Proof of Theorem 3.1 Let ao be the constant from Proposition 3.2. Fix ao < ao/2 to be defined later. Let F b e a quotient space of X, and let || • ||2 be a Euclidean norm on Y such that dim Y = a^n and (3.6), (3.7) and (3.8) are satisfied. Fix an integer ra with 2aon < m < aon to be defined later. Let 0 < a < a 0 . Assume that for every m-dimensional quotient Y\ of F , condition (i) is violated. For a given Y\ and a fixed e > 0 small enough this means the following. In case of the distance estimate, the measure of the set of pairs (Fi, F2) which satisfy (3.3), is smaller than 1 — e n ; in case of the basis constant estimate, for ra' = 5cm/4 and P m / being the orthogonal projection on the first ra' coordinates, the measure of the set of quotients F ' s of Pm>(Yi) which satisfy (3.4), is smaller than 1 — e n . By the concentration of measure phenomenon, for any Y\ we have: either a random pair (Fi,F 2 ) of cmdimensional quotients of Yi actually satisfy
(3.15) or a random cm-dimensional quotient F of Pm/(Yi) satisfies bc{F)
(3.16)
We will show that there exist an ra-dimensional quotient X\ of Y for which (ii) holds. Define a' by a'm = an. Set /?' = a ; / 4 and k = 3/3'm/4. Now the proof is similar as in [M-T] Theorem 5.3, in which the use of Szarek's volume ratio argument is replaced by Proposition 3.3. It shows that if any of (3.15) or (3.16) holds then every ra-dimensional subspace Z = (Yi)* of Y* contains a subspace H with dim if = /?'ra/2 such that
214
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
In general, a,\ depends on a', /?' and k/fl'm. However, as we have a! < 1/2, /?' = (l/2)(a'/2) and k < (3/4)/?'m, then ax may be taken as an absolute constant. By Proposition 5.5 from [M-T], this implies the existence of a subspace 5 of Y* such that (3.17) and k = dim 5 > dim Y* — m = aon — m. We want to show that on a certain m-dimensional subspace of 5, the original norm and | • I2 are equivalent, with proper constants. If a\ > 1, let s be the smallest integer larger than log 2 ai. Applying Lemma 3.4 s times we get a subspace Z of 5, with dimZ > 0.015(aon — m) such that d(Z, Iftmz) < K* ^ *s well-known that given two ellipsoids on an /-dimensional space, there exists a subspace of dimension 1/2 on which the ellipsoids are proportional. Comparing a distance ellipsoid on Z with the ellipsoid defined by | • | 2 , and passing to a (dim Z/2)-dimensional subspace of Z if neccessary, we get a subspace Z\ C Z on which (3.5) holds. Let m be the largest integer such that m < 0.0V(aon — ra)/2 and set ao = 0.01aao/2(l + 0.015). Since dimZx > m, Xx = Z\ is the required quotient of -X". If ai < 1, the proof can be completed by applying the latter step alone. • The following corollary is very close to Theorem 3.1, and it is its immediate consequence. Corollary 3.5 There is a constant 0 < a0 < 1/2 such that the following holds, for every 0 < a < ao. Let X be an n-dimensional Banach space, and let \ • | 2 be a Euclidean norm on X satisfying (S.I) and (3.2). There is an integer m with 2aon < m < n, and an m-dimensional quotient Xx of X such that whenever d(Xi,l^n) > 10 then a random pair (i^,!^) of an-dimensional quotients of X\ satisfies d(FuF2)>d(X1,l2J2
(3.18)
and a random an-dimensional quotient F of X\ satisfies bc(F)>d(Xt,O.
(3.19)
Mankiewicz & Tomezak-Jagermann: Pathological properties and dichotomies 215
If | • 12 is the norm determined by the ellipsoid of minimal volume containing the unit ball of X, then a version of the corollary, with additional universal constants smaller than 1 on the right hand sides of (3.18) and (3.19), is satisfied for an arbitrary a < 1/2 and any m with a < m < 1/2 (c/. Theorem 5.3 in [M-T]). Another consequence of Theorem 3.1 is the following infinite-dimensional result, rather weaker than the theorem itself. It is more natural to state it for subspaces of a given space rather than for quotients. Corollary 3.6 There is a constant 0 < ao < 1 such that the following holds, for every 0 < a < ao- Let Z be a Banach space. Let f : N —> i?+ be a non-decreasing function, lim/(n) = oo. Exactly one of the following two conditions holds. (i) There exist a sequence {Zn} of subspaces of Z, with kn = dimZ n —• oo, as n —> oo, and Euclidean norms \ • \n on the Zn 's so that the following holds. There is a family T1 of random pairs (E\,E2) of akn-dimensional subspaces of Znj such that
inf
d(£?,,^)//(fc B )->oo
(3.20)
\EE)GT'
and there is a family J~ of random akn-dimensional subspaces E of Zn such that ^cx>. (3.21) (ii) For every k-dimensional subspace {Z'} of Z, and every Euclidean norm \ • | on Z', there is a family T of random ak-dimensional subspaces E of Z1 such that d(E,l2ak)
216
Mankiewicz & Tomczak-Jagermann: Pathological properties and dichotomies
where a > 1 is a universal
constant.
References [B-S]
BOURGAIN, J . &: SZAREK, S. J., The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization. Israel J. of Math. 62 (1988), 169-180.
[B-T]
BOURGAIN, J . & TZAFRIRI, L., Invertibility of "large" submatrices with applications to the geometry of Banach spaces and harmonic analysis. Israel J. of Math. 57 (1987), 137-224.
[H]
HADWIGER, H., Vorlesungen liber Isoperimetrie. Springer Verlag, 1957.
[F-L-M]
FlGlEL, T . , INDENSTRAUSS, J., &: MlLMAN, V. D., The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), 56-94.
[M-T]
MANKIEWICZ,
P.
Inhalt:
Oberflache
& TOMCZAK-JAEGERMANN,
subspaces and quotients of finite-dimensional Preprint, Odense University, 1989.
N.,
und
Random
Banach spaces.
[M.I]
MlLMAN, V . D., Inegalite de Brunn-Minkowski inverse et applications a le theorie local des espaces normes. C. R. Acad. Sci. Paris., 302 Ser 1.(1986), 25-28.
[M.2]
MlLMAN, V . D., Almost Euclidean quotient spaces of subspaces of finite dimensional normed spaces. Proc. Amer. Math. Soc, 94 (1985), 445-449.
[M.3]
MlLMAN, V. D., A new proof of the theorem of A. Dvoretzky on sections of convex bodies. Func. Anal. AppL, 5 (1971), 28-37. (translated from Russian).
[M-P]
MlLMAN, V. D. k PlSIER, G., Gaussian processes and mixed volumes. Ann. ofProb., 15 (1987), 292-304.
Mankiewicz & Tomezak-Jagermann: Pathological properties and dichotomies 217
[M-S]
MlLMAN, V . D . & SCHECHTMAN, G., Asymptotic
theory
of fi-
nite dimensional normed spaces. Springer Lecture Notes No. 1200 (1986). [P-T]
PAJOR, A. & TOMCZAK-JAEGERMANN, N., Volume ratio and other s-numbers of operators related to local properties of Banach spaces. J. offline. Anal., 87 (1989), 273-293.
[P]
PlSIER, G., Volumes of Convex Bodies and Banach Spaces Geometry. Cambridge Univ. Press, 1989
[Sz]
SZAREK, S. J., The finite dimensional basis problem with an appendix on nets of Grassmann manifolds. Acta Math., 151 (1983), 153-179.
[T]
TOMCZAK-JAEGERMANN, N., Banach-Mazur Distances and Finite Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific Sz Technical, Harlow and John Wiley, New York, 1989.
A NOTE ON A LOW M*-ESTIMATE V. Milman* Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv, Israel
1. Introduction and Notations Consider a finite dimensional normed space X = (R n , || • |,| • |) equipped with a norm || • || and a euclidean norm | • |. Then the dual space X* = (R n ,|| • ||*) equipped with the dual norm ||x||* = sup{|(.r,2/)| | \\y\\ < l } arises naturally (here (x,y) is the inner product which produces the euclidean norm \x\2 = (x,x)). Let S = {x € R n , \x\ = 1} be the euclidean sphere equipped with the rotation invariant probability measure fi. The following two parameters play an important role in the study of the linear structure of the space X: M(=MX)=
f
||*||d/i(s) and M*(= Mx.) = [
\\x\\mdfi(x) .
The discussion in this note will center around the following (known) proposition. Main Proposition 1. There is a function /(A), 0 < A < 1, such that for all integers n and k < n and any X = (R n ,|| • ||,| • |) there is a subspace Ek of codimEk = k (i.e., Ek G Gntn-k - the Grassmann manifold of(n — k)-dimensional subspaces ofRn) and for every x £ Eh J
-$r\*\ < Ikll •
(i)
This proposition was presented for the first time at the Laurent Schwartz Colloquium (Summer 1983) where (see [Ml]) it was shown that it plays a central role in the developing of a "proportional" theory (as it was called there). Since that time it has been used in different forms in most publications on Local Theory (in the study of finite dimensional spaces or operators). In the above form (1), different estimates on a function /(A) may be important and we will discuss this below. However, even the roughest and easiest estimate with an exponential dependence on A (for A —»• 0) * Supported in part by an G.I.F. Research Grant
220
Milman: A note on a low M*-estimate
already had unexpected and, probably, the most interesting applications, as can be seen in [M2]. In addition, a completely different form of inequality (1) with the spherical (Gaussian type) average M* substituted by Rademacher type average (i.e. average over signs), was suggested by the author (unpublished) and used very successfully in [B.Tz]. Today, a few different approaches to the Proposition are known, leading to different functions /(A): (i) the original approach (1983) used Urysohn's inequality (see [Ml]) and gave /(A) = cxfx for a number 0 < c < 1; (ii) in the next step (1984), we returned to the use of isoperimetric inequality on the euclidean sphere Sn to show that (see [M3]) /(A) > cX . (It was later shown ([MP], Appendix) that this is an easy consequence of pure entropy considerations.) Moreover, approach [M3] gives /(A) > A + o ( l - A) when A -> 1 . (We note this because inequality lim /(A)/A > 1 was recently used.) A->1
(iii) Asymptotically (A —r 0) the best dependence /(A) > c\/A was received by Pajor-Tomczak [PT] in 85-86 by adapting Gluskin's method for £" to a general normed space and joining it with Sudakov minoration estimate on covering numbers, (iv) Y. Gordon [G] showed, using his minimax comparison principle, that, in fact, for 0 < A < 1,
(i.e., a constant c above may be taken ~ 1). In this note, we give another proof of the Proposition with an estimate (iii) as in PajorTomczak. For this purpose, we adapt to a general normed space a new approach of Makovoz [Ma] to prove Garnaev-Gluskin's estimates on the relative position of the Euclidean ball of £f in the unit ball of tf. I would like to emphasize that I found the Makovoz arguments very interesting (they are, in fact, a "right" organization of Garnaev-Gluskin's arguments) and a way of adapting them to a general normed space could be of independent interest. Some new results in this direction are presented in section 4.
Milman: A note on a low M*-estimate
General Remark.
221
All proofs of Proposition 1 have a probabilistic nature. Therefore, in fact,
we obtain a set A C Gn>n_fc of subspaces which satisfy (1) of "almost" full measure. We mean by this that f.i(A) > 1 — a(n,A), where a(n, A) —• 0 (n —> oo) and A > fixed e > 0; here \i is the Haar probability measure on the Grassmann manifold Gn>n_fc. It is exceptable jargon to call an individual element of A a "random" subspace and to state Proposition 1 for a "random" fc-codimensional subspace. Usually, we automatically derive an estimate on a(n, A) which, in all cases, has a form a(7i,A)
assume that 0 GA".
222
Milman: A note on a low Af""-estimate
Acknowledgement.
I am grateful to Professor G.G. Lorentz who brought my attention to
Makovoz's paper. I would like to thank IHES where this paper was written.
2. Preliminary Lemmas The following two lemmas are close enough to well-known results. So we will not develop detailed proofs. We denote below by P^ the orthogonal projection onto subspace f. Lemma 2.1. Let \x\ < \\x\\ (i.e., K C D). Then for every integer 1 < k < n, for a "random" subspace £ € Gn>k (of dimension k)
( Cy/kJnD(() if y/kj^ > M* K CI [CM*D(t) (here, as usual, C is a universal constant). Proof:
First consider a case y/k/n ~ M*, i.e., k ~ (M*)2n.
The standard estimate of the
dimension of euclidean sections in Dvoretzky Theorem (see [M.Sch], 4.2) for the dual space X* shows that, for a "random" subspace f, P±K ~ M*D(£). Then of course, the same is true for k < n(M*)2 because P^D(r/) = D(£) for f C 77 (f and 77 being subspaces of R n ). In the case of k > n(M*)2, note that we want to prove now only embedding P$K C Cy/k/nD(£) (in the previous case we had, in fact, equivalence). For the dual sets, this means the following inequality for norms
11*11* < • The standard "concentration phenomenon" approach to Dvoretzky's Theorem (as described in [M.Sch], Chapters 2-4) gives this upper bound for a random ^-dimensional subspace of A"* (under c o n d i t i o n \\x\\~ < \ x \ ) .
Remarks. 1. I realize that the above sketch of the proof may suffice only for readers well versed with the methods of the first part of the book [M.Sch]. 2. The interesting point of Lemma 2.1 is that it combined the modern form of Dvoretsky's Theorem (as was put forward in [M4] and later developed in [FLM]) and Johnson-Lindenstrauss' Lemma. [J-L] which states:
Milman: A note on a low M*-estimate
223
Let A C Sn be a set of at most 2k points (i.e., #A < 2k) on the euclidean sphere Sn; then for a random orthogonal projection P{ onto a ^-dimensional subspace f for any x€ A
We derive from Lemma 2.1 only an upper bound: \P{x\ < Cy/k/n.
Let K = Conv A U (—A)
and X = XK- Then M* of A' is at most Cy/k/n (for a universal constant C) which implies the upper bound P^K C Cy/k/nD(£). Lemma 2.2. There are constants C and c > 0 such that for every integer k, logra < k < n, there is a set Ak C .S*"1"1, #Ak = Ck, producing the body Tk = ConvAfc U (~Ak) with the following property: D for the orthogonal projection P$ onto a random k-dimensiona,l subspa,ce £ C Rn . Sketch.
The Lemma may be verified by a probabilistic argument (althoug I would prefer to see
here the construction of a set Ak)- For that, note that for a set T = {xt-} C 5*fc~l, # T = 10fc, of randomly and indipendently chosen points {x^ on the sphere and any numbers a*, \ < a< < 2, with high probability Conv{±a»Xj} 3 cD for some universal constant c > 0. Combine this observation with Johnson-Lindenstrauss' lemma from Remark 2 above. Normed spaces Xk = Xxh give a certain scaling which controls, as we will see, the
Remark 3.
behavior of a general normed space. Note that spaces I™ and t1^ belong to this scale: 0% = X\o% n and £^ = Xcn (the last fact follows from Kasin's well-known result [K] stating that a random orthogonal projection of a ??-dimensional cube onto a y-dimensional subspace is uniformly isomorphic to a euclidean ball). Problem.
A general problem which arises at this point is to construct, for every n > k > log n,
a. symmetric set Ak, #Ak = 2fc, such that for a set of ^-dimensional subspaces 21 C Gn%k of measure > j^jr (so, very small but anyway not too small) Pi(CcmvAk)D6D(0 for a maximal possible 6 > 0.
(V ( € 21)
224
Milman: A note on a low M*-estimate
It is, of course, enough for applications to prove an existence of a set Ak which gives a large enough S. But how large can this 6 be? As stated in Lemma 2.2, a set 21 of subspaces is of almost full measure if S ~ y/k/n. Remark 4.
Could this be improved for a set of small measure?
For a small S ~ J^/ J\og 2
Take integer t such that [ / l o g ^ ] = k > logn and consider the set 01 C 5 n ~ 1 of all ndimensional vectors with 0 and ± 4 - enters (so, O's appear exactly n — I times). Then #01 = 2* (7) ~ 2'lo« ^
- 2k. Consider T = Convtfl. Then ^D
C T which means that
So, we don't need to take any projections on subspaces at all.
3. Proof of Proposition 1 We will now prove Proposition 1 with estimate iii) using some ideas from Makovoz paper [Ma]. Lety = ( y 1 , . . . , y f c ) c 5 n " 1 . Consider
Lemma 3.1 [Ma]. Denote by P the probability porduct-measure on the product of k spheres k
•Q gn-i (and n
x£5 "
(Therefore, ^
every
sphere equipped with the rotation invariant probability measure). For every
1
for any set 0 1 C R n , # 0 1 < e f c / 2 , there
isy = (yu...
,yk) such
that
1(
J Q ^ < F(x,y)
<
for any x € 0 1 . )
Lemma 3 may be proved by direct computation (see [Ma]). In the framework of Local Theory, it is an immediate consequence of Bernstein's inequality (see, e.g. [H] or [BLM] for exactly such a use): consider a function
= e* — 1). Therefore, by the standard sampling method for V>2-distribution (see [BLM],
Prop. 1, (iii)) k
Prob [ye
E[ 5 - 1
«*)\ - j -
Milman: A note on a low M*-estimate
225
Take e ~ £o/y/n. and we derive Lemma 3.1 with some constants C\ and C2 instead of 100 and 3 above. Remarks. 1. If we are interested only in an upper estimate like F(x,y) < ^jl^l (for & large T), we can take e = -j£ to be large and improve the estimate of the probability of t/'s. 2. Taking T = c^Jn/k, we have Prob
"
1
and, therefore, we may satisfy the inequality
F(x,y)
Vx G £. AT
We use a notion of the covering number N(K, D) = min {N \ 3{.T,}{^ such that A' C U(a!t + £))}• l
It is well-known (Sudakov's inequality - see, e.g. [LT]) that (A' below is the unit ball Kx of X)
for our normalization. Let O^o = {^t} be a D-net of A' such that #91 < ek and let
cn = m0 u Akl where Akl is a set on S"" 1 from Lemma 2.2 (and ki = cxk will be chosen next). Therefore, #01 < Ck for a universal constant C. Use Lemma 3.1 for ki = C\k (ci = 21ogC - universal constant).
Then there is y =
(Vi € .S'"" 1 )^! such that for every Xj 6 01
)~&L.
(3.1)
226
Milman: A note on a low M*-estimate
Moreover, we have a large measure of such {#»} € II 5 n ~ 1 which means a large measure of subspaces By Lemma 2.1 (which we apply to a body K H D noting that M*(K H D) < M*(K) =
y/k/n),
there is a set of large measure of &i -dimensional subspaces such that the orthogonal projection on any £ from this set P^(K fl D) C c y ~ • D(f )• We now combine all 3 conditions (from Lemmas 2.1, 2.2 and 3.1) and find a subspace £ G Gnikl from Lemma 2.1 for the set K C\ D, and, at the same time, from Lemma 2.2 for ki, and f = spai^y,)^! satisfying (3.1). We put (briefly) F(x) = JP(X,J/). Consider E = £'L(= {2/t}'L)- We will show that E is a subspace which we are looking for. Clearly codimi? = k\ ~ fc and F{x) = 0 for any x € -E. We have to show that for every x 6 £7,
Nl < i. |z| < Const.(?) . Write x^= x" + ar' where x" 6 ^o and |.r'| < 1. We wiU estimate |x"| from above by estimating F(x") ~ ^ , Because our x <E K H (x" + Z)) we have x ; = x - x" € 2(A' fl D) (note that K n (ar" + D) C a:" + (2A') H J9 C a" + 2(/f H D)). By Lemma 2.1, for some constant c 2 ,
Then there exists x[ € 2[/v D D], P^a;1 = P^a^, and CZX'Q € Conv{zi}XieAki
= Tkl
(again, for some universal constant cz > 0). It shows that
F(x'0)< -i-maxP(^)- 4 = (recall that Zj e Ti and therefore F(zj) ~ ^ 1 ) . It remains to note that F(x") = F(x'Q) < c/y/n which implies |ar"| < Const, and |.r| < |x"| + |ar'|.
a
4. Some New Connected Results and Problems In fact, a stronger statement was proved above. First, define entropy numbers (the inverse function to covering numbers): for every integer k > 1, ek = ek(K,D) = inf {e \ N(K,sD) < 2^}
.
Milman: A note on a low M*-estimate
227
Assume that e* = 1. Then it was proved that, for a random subspace Ek of codim £ fc ~ k and any xeEk,
Note, that the expression M*(KC\ekD) is independent of homothetic change of the euclidean norm. Therefore we obtain Proposition 2. For any X = (Rn, || • ||,| • |) and any integer k < n for a random subspace Ek of codim Ek = k
where c is a universal constant. The above inequality is stronger than Proposition 1 with /(A) given by iii), because m
M (K n ekD) < M*(K). For the following result we introduce the so called "Sudakov" numbers. Let K be, as above, the unit ball of a space X — (Rn, || • ||) and let e» be the entropy numbers of K. Consider
pVJ i By Sudakov's minoration theorem S(K) ^ y/nM*. Therefore, to substitute inequality (1) in the framework of Proposition 1 by the following one
would be an improvement over all known estimates. Unfortunately, we cannot prove this fact but will prove one close to it. Theorem 3. For any X = (R n , || • ||,| • |) and any integer k < n for a random subspace Ek of codim Ek = k
(c is a universal constant). Proof:
Let % be the smallest e i+1 -net of K. Then #9V,- = 2j. Let ki be such that eki = ^.
Then, by Remark 1 to Lemma 3.1. we have a large measure of jTs (i.e., of spanp"= f G Gn,k) such that (4.1)
228
Milman: A note on a low M*-estimate
for any x € 9^*- — (HJ^J. We achieve this first for a fixed i but choose next the same |/0 for all i, 0 < i < t. (We choose later t ~ \ogn/k.) As in the proof of Proposition 1 a subspace which we are looking for is Ek = [fo]-1. To prove this, take any x G Ek, \\x\\ < 1, and decompose it (for every z)
where x? e mki and |*J| < l/2 f . Then x? - x? +1 = x; +1 - xj and |x; +1 - xj| < ^ Because |x| < |xi'| + |.TJ| < 1 + |xi'| we have to estimate |xi'| ~ y/nF(x^yo).
+ ^r = jAr. Again, we will
abbreviate F(z,lj0) to JP(^r). Note that F is a seminorm and F(x) = 0, so
1 t-1 1
(use Remark 2 of Lemma. 3.1 to estimate the last term) ^
Choose now / ~ log ^. Then v'Tl^tl ^ 1 a n d we can continue the above inequality
Remark.
Note an easy improvement of the inequality in Theorem 3: we can substitute S(K)
by Sk(K) = sup y/j€j . >fc
Milman: A note on a low M*-estimate
229
References [BLM]
J. Bourgain, J. Lindenstrauss, V. Milman, Minkowski sums and symmetrizations, Springer-
[BTz]
J. Bourgain, L. Tzafriri, Invertibility of large submatrices with applications to geometry
Verlag, Lecture Notes in Mathematics, 1317 (1988), 44-66.
of Banach spaces and harmonic analysis, Isr. J. Math. 57 (1987), 137-224. [FLM]
T. Figiel, J. Lindens trauss, V. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 129 91977), 53-94.
[G]
Y. Gordon, On Milman's inequality and random subspaces which escape through a mesh in R n , Springer-Verlag, Lecture Notes in Mathematics 1317 (1988), 84-106.
[H]
W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Stat. Assoc. 58 (1963), 13-30.
[JL]
W.B. Johnson, L. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Proc. Conf. in Honour of S.Katutani.
[K]
B.S. Kasin, Sections of some finite dimensional sets and classes of smooth functions, Izv. ANSSSR, ser. mat. 41 (1977), 334-351 (Russian).
[LT]
M. Ledoux, M. Talagrand, Isoperimetry and processes in probability in Banach spaces, Springer-Verlag (1990).
[Ma]
Y. Makovoz, A simple proof of an inequality in the thoery of n-widths, Constructive Theory of Functions '87, Sofia (1988).
[Ml]
V. Milman, Geometrical inequalities and mixed volume in Local Theory of Banach Spaces, Asterisque 131 (1985), 373-400.
[M2]
V. Milman, Volume approach and iteration procedures, in Local Theory of Normed Spaces, Springer-Verlag, Lecture Notes in Mathematics 1166 (1985), 99-105.
[M3]
V. Milman, Random subspaces of proportional dimension of finite dimensional normed spaces; approach through the isoperimetric inequality, Springer-Verlag, Lecture Notes in Mathematics 1166 (1985), 106-115.
[M4]
V. Milman, A new proof of the theorem of Dvoretzky on sections of convex bodies, Funkcional. Anal i Prolozen 5 (1971), 28-37 (Russian).
[MP]
V. Milman, G. Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54, No. 2 (1986), 139-158.
[MSch]
V. Milman, G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Springer-Verlag, Lecture Notes in Mathematics 1200 (1986), 156pp.
[PT]
A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of finite dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 637-642.
The pp in Pisier's Factorization Theorem S.J. Montgomery-Smith Abstract We show that the constants in Pisier's factorization theorem for (p, l)-summing operators from C(H) cannot be improved.
A theorem of Pisier (see [3]) states the following. Theorem 1. Let T : C(ft) —» X be a bounded linear operator, where £1 is a compact Hausdorff topological space and X is a Banach space. Then the following are equivalent. i) T is (p, 1)-summing. ii) There is a constant C < oo and a Radon probability measure fi on £1 such that for all f e C(H) we have
P7II < c \\f\\iiM \\fC*.
(i)
Hi) There is a constant C < oo and a Radon probability measure fi on Q, such that for all f £ C(H) we iiave l|r/||
232
Montgomery-Smith: The p1!P in Pisier's factorization theorem
Theorem 3. Given t > 0, there is an operator T : C(Q) —> X such that for any Radon probability measure \i on Q,, if C is the least number satisfying (1) or (2), then C >
Construction. Let 1 < S < N be integers, and let Ct be the collection of 5-subsets of {1,2,..., N}. For each 1 < n < N, let On = {u G ft : n G a;}. We note the following facts for later on:
N
We give Q the discrete topology, and define a norm || • ||+ on C(O) by
We let T be the canonical embedding
Lemma 4. For 1 < p < oo, the (p, l)-summinff norm ofT may be estimated by
(T) <
Ns~1+7
In order to show Lemma 4, we will need two more lemmas. Lemma 5. If 1 < p < oo, and T : C(£l) —> X is a bounded linear operator, where X is a Banach space, then the (p, 1)-summing norm ofT may be calculated by the formula
where the supremum is over all sequences fi, / 2 , . . . , fs of disjoint elements of the unit ballofC(Q). Proof.
See [2], Lemma 6 or [1], Proposition 14.4.
•
Montgomery-Smith: The plfP in Pisier's factorization theorem
233
Lemma 6. If 1 < n < TV, then
Proof.
A simple counting argument shows that |Qi u n2 u . . . u o m \ Hi u o 2 u . . . u <
and this is bounded by (N - m ) 5 " 1 / ^ - 1)!. Proof of Lemma 4.
•
By Lemma 5, it is easy to see that
r,tl{T) = sup I (JT \\XBjA ' 1 = sup H £ |Bn|'J ' 1 , where the supremum is over disjoint sets I?i, B2,..., 5;v ^ ^ such that J5n C Q,n for each 1 < n < N. Since Oi, O2,..., 0// interact with one another in a completely symmetric fashion, we may assume, without loss of generality, that \Bi\ > \B2\ > ... > \B^\. Now N
N
n=l
n=l
/
n
\Tn=l
p l
(We take BN+i = 0.) Since \Bn\ ~ - \Bn+l \p~l > 0, and B1 U B2 U ... U Bn C Oi Uft2U . . . U O n , we have, by Lemma 6, that
E i5«ip ^ TS^TV E ( n=l
\m=l
Now, applying Holder's inequality and dividing, we deduce
234
Montgomery-Smith: The p1!P in Pisier's factorization theorem
Finally, we estimate the last quantity by an integral, and derive
( S - l ) ! ( 5 p - P + l)7 as desired.
D
Proof of Theorem 3. By the hypothesis on C, there is a probability measure fj, on Q such that inequality (1) holds. In particular, if we subsitute / = Xftn, w e deduce that
Hence J
n=l
n=l
and so by equalities (3) and (4) we have
C>
Thus, by Lemma 4, we deduce
Choosing N much larger than 5, we find that
Finally, choosing S large, we have the desired result, that is, C > p~p
TTPJI(T) (1 —
e).
•
Montgomery-Smith: The plfP in Pisier's factorization theorem
References 1.
Jameson G.J.O.: Summing and Nuclear Norms in Banach Space Theory. London Math. Soc, Student Texts 8, 1987.
2.
Maurey B.: Type et cotype dans les espaces munis de structures locales inconditionelles, Exposes 24^25. In: Seminaire Maurey-Schwartz 1973-74 (Ecole Poly technique). Pisier G.: Factorization of operators through Lpoo or Lv\ and non-commutative generalizations. Math. Ann. 276 105-136 (1986).
3.
S.J. Montgomery-Smith, Department of Mathematics, University of Missouri at Columbia, Columbia, Missouri 65211, U.S.A.
235
Almost differentiability of convex functions in Banach spaces and determination of measures by their values on balls* D. Preiss
Abstract. We prove a new differentiability property of continuous convex functions on separable Banach spaces. As an application we give a new proof of the result of [2] that Borel measures on separable Banach spaces are determined by their values on balls. Let / be a convex continuous function defined on an open convex subset G of a real Banach space E. We recall that the subdifferential of / at a point x G E is the set df{x) of all (necessarily continuous) linear functionals x* such that f(x + h) — f(x) > (A, #*) for every h G G. Using the fact that / is convex and continuous, we easily see that the set T ( / , x) := {h G E; /'(#, h) = —/'(#, —h)} is a closed linear subspace of E. (Recall that f'(x,h) = limt\o(f(x + th) —f(x))/t.) Moreover, the Hahn-Banach theorem immediately implies that T ( / , z ) = n{Ker(z* - y*); z*,y* £ df(x)}. Clearly, the function / at the point x is Gateaux differentiate in the direction of the subspace T ( / , x) and is nondifferentiable in any direction not belonging to T(/, x). Hence the codimension of T(f,x) (which will be denoted as dim(df(x)) (and called the dimension of the subdifferential off at x) measures how much / is Gateaux nondifferentiable at x. Clearly, Gateaux differentiability of / at x is equivalent to dim(df(x)) = 0. For separable spaces, the classical result of Mazur according to which / is Gateaux differentiate at the points of a residual subset of G also implies -Classification: Primary 46G05. Secondary 28C15, 46G12.
238
Preiss: Almost differentiablity of convex functions
that for every x 6 G df(x) = p | w*-closed convex hull {/'(*); \\z - x\\ < r, f'(z) exists}. r>0
The proof of the Mazur theorem and other relevant information can be found in [1], A more general form of the above corollary of the Mazur theorem is given in Proposition 1.1 below. Motivated by an application to the question of determination of measures by their values on balls, we intend to replace in this statement wu;*-closed convex hull" by "t/;*-closure". This needs, of course, a weakening of the requirement of Gateaux differentiability. (Just consider the absolute value function on R.) However, it turns out that for our application, Gateaux differentiability in the direction of some finite codimensional linear subspace suffices. It is our aim to prove that, with these two replacements, the expression for the subdifferential remains true (see 1.4). This result is used to give a new proof of the result of D. Preiss and J. Tiser [2] that Borel measures on separable Banach spaces are uniquely determined by their values on balls. Our main result is based on a new estimate of smallness of the sets {x 6 G\ dim(9/(x)) > k} for convex continuous functions / . (Theorem 1.3.) Let us remark that L. Zajicek [3] proved that a subset of G is of such a form if and only if it can be covered by count ably many k + 1 codimensional 6- convex (or (c — c)-convex) surfaces. (Since we will not use this fact, we refer the reader to [3] for the definition.) However, our result does not seem to follow directly from this interesting characterization of the smallness of these sets. Except for the notation introduced above, we shall also denote by B(x, r) and B°(x, r) the closed and open ball centered at x with radius r, respectively.
1. Almost differentiability of convex functions 1.1. Proposition. Suppose that f is a convex continuous function defined on an open convex subset G of a real Banach space E and that, for each sufficiently small r > 0, Dr is a subset of\J{df(z); z £ B°(x,r)} such that the set {z e B°(x,r); Dr 0 df(z) f 0} is dense in B°(x,r). Then df(x) = p | u;*-closed convex hull Dr. r>0
Preiss: Almost differcntiablity of convex functions
239
Proof. Since the mapping x \-> df{x) has u;*-compact convex values and is it;*-upper semicontinuous,
df(x)
=> fl ^*-dosed convex hull \J{df(z)', z € B°(x, r)} r>0
D P | u;*-closed convex hull Dr. r>0
To prove the opposite inclusion, suppose, to the contrary, that x* £ df(x)\M where M = (\>o w*-closed convex hull Dr. Then the Hahn-Banach theorem provides us with h G E such that \\h\\ = 1 and (h,x*) > sup{(/&,z*); z* £ M}. Let c, d be real numbers such that (h,x*) > c> d > sup{(/i, z*); z* £ M}. For every r > 0 such that B°(x, r) C G and the set {z £ B°(x, r); Dr fl df(z) ^ 0} is dense in B°(x,r) we find e € (0,r/4) such that |/(w) - / ( x ) | < (c-d)r/8and |/(u+r/i/4)-/(a;+r/i/4)| < (c-d)r/8 whenever u e B°(x,e). Then for each u G B°(x,e) the function g:t e (-3r/4,3r/4) h^ f(u + th) fulfils ^(r/4)-(/(O) = f(u + rh/4)-f(u) > f(x + rh/4)-f{x)-(c-d)r/4 > ((hj x*) — (c — d))r/4 > dr/4. Since g is convex, it follows that g(t) — g(s) > d(t — s) whenever r/4 < 3 < t < 3r/4. Consequently, (A, z*) > c? whenever z * € U{^/(^); Ik "" (x + r ^ / 2 ) | | < ^}- Since our assumption implies that Dr fl \S{df{z); \\z - (x + rh/2)\\ < e} ^ 0, there are < € Z?r such that (hiz*) ^ ^- Finally, using the fact that limsupr \^ 0 \\z*\\ < oo, we conclude that the set of all ^-accumulation points of {z*} for r \ 0 is a nonempty subset of M. But, since each z* € M fulfils (A, z*) < d, this is a contradiction. 1.2. Proposition. Le£ f be a convex continuous function defined on an open convex subset G of a separable real Banach space E and let x £ G. Then the following two statements hold. (i) The set of all x* € df(x) such that f\x,h) T(/, x) is norm dense in df(x).
> {h,x*) for every h £
(ii) If dim(df(x)) < oo then for any x* with the property from (i) there is rj>0 such that f(x + h) - f(x) > (h, x*) + rj dist(fc, T(/, x)) for every
heE. Proof. Since E is separable there is a sequence z\, z\,... 6 df(x) such that T(/,x) = n £ i fljLi Ker(zf - zj). Hence, to prove the first statement,
240
Preiss: Almost differentiablity of convex functions
it suffices to consider for any x* £ df(x) the sequence x* := (1 — 2~q)x* + To prove the second statement in case dim(df(x)) > 0 (if dim(df(x)) = 0 then T(fjX) = E and the statement is obvious), we first use the facts that the function h h-» f'(x,h) — (h,x*) is convex, continuous, and positively homogeneous and that it equals zero on T(/, x) to infer that it is of the form g o /c, where g is a convex, continuous, and positively homogeneous function on the factor space E/T(f, x) and K is the canonical projection. Since our condition upon x* implies that g is positive off the origin, we deduce from the compactness of the unit sphere in E/T(f, x) that there is rj > 0 such that, g(u) > rj\\u\\ for every u eE/T(f,x). Hence f(x + h) - f(x) - (h,x*) > f\x,h) - (h,x*) > fi\\K(h)\\ =rfdist(h,T(f,x)) for every h e E. 1.3. Theorem. Suppose that f is a convex continuous function defined on an open convex subset G of a separable real Banach space E, k = 0,1,..., and that TT is a linear map of E onto a real Banach space F such that dim(Ker(7r)) < k. Then the ir image of the set {x 6 E; dim(df(x)) > k} is a first category subset of F. Proof. Let S be a countable dense subset of E and let V be the family of all pairs (V, e*) where V is a k +1 dimensional subspace of E generated by a finite subset A of S and e* is a real valued linear functional on V attaining only rational values on A. First we observe that for every x G E with dim(df(x)) > k there is a pair (V, e*) £ V and 77 > 0 such that for every h £ V. Indeed, since the subspace T(/, x) of E has codimension at least k + 1, there is a k + 1 dimensional subspace V of E spanned by a finite subset of S such that V D T(f,x) = {0}. Using first 1.2(i) and then 1.2(ii), we find x* € df(x) and 77 > 0 such that f(x + h) - f(x) > (h,x*) + 2rf\\h\\ for every h 6 V. Now it suffices to find a linearly independent subset A of V fl S and to define e* so that its values on A are rational and sufficiently close to the values of x*. Since the family V is countable, our statement will be proved by showing that for any (V, e*) G V and for any 7/ > 0 the TT image of the set M:={xe
G; f(x + h) - f(x) > (/*, c*> + 9 ||A|| for every
heV}
Preiss: Almost differcntiablity of convex functions
241
is a first category subset of F. Let W be a subspace of V such that V = W 0 {V D Ker(Tr)) and let U be a closed subspace of E containing W such that U 0 Ker(?r) = E. Also, let X be a subspace of Ker(?r) such that Ker(?r) = X 0 (V C\ Ker(Tr)). Since the restriction of ir to U is a linear isomorphism, we may identify F with U'. The map TT then becomes the projection of E onto U along Ker(Tr). We also observe that dim(X) < dim(W), since dim(X) = dim(Ker(7r)) — dim(V 0 Ker(Tr)) < Jfc-dim(V flKer(Tr)) and dim(W) = dim(V) -dim(VnKer(7r)) = ifc + l - d i m ( V n K e r ( 7 r ) ) . Since / is convex and continuous on G, each point of G has a neighbourhood on which / is Lipschitz. Consequently, it suffices to show that the 7T projection N of the set M 0 B(xo,ro) is a nowhere dense subset of U whenever / is Lipschitz on B(xo,3ro) C G with Lipschitz constant, say, a. Also, since the spaces W and V C\ Ker(Tr) are finite dimensional and since W D [V f) Ker(?r)] = {0}, there is a constant b G (0, oo) such that IIw|| < b\\w + v|| whenever w G W and v G V D Ker(7r). Whenever Uj —> u is a convergent sequence of elements of N (considered as a subset of [/), we find ZJ G Ker(?r) such that Uj + Zj G Mfl-B(x 0 , r 0 ). From E = [/0Ker(7r) we see that the sequence Zj is bounded. Since Ker(Tr) is finite dimensional, we may also assume that Zj converges to some element, say 2, of Ker(7r). But then the definition of M implies that u + z G M n B(x0, r 0 ), which shows that u G N. Consequently, TV is a closed subset of U. Finally, to prove that N has empty interior, we show that for each w G U the set Nf](w+W) has empty interior in w+W: Let Ui, ti2 G NC\(w+W). We find ui, v 2 G VnKer(Tr) and Xi,x 2 G X such that Ui + Vi+Xi G Mfl-B(a:o,ro) and u 2 + v2 + x2 G M n ,B(a:o,ro). Then the definition of M implies / ( u ! + «! + xa + h) - / ( m +
Vl
+ *i) > (fc,c*) + rj\\h\\
and / ( u 2 + v2 + a:2 + A) - f(u2 + v2 + x2) > (fc,c*) +
T/||/I||
for every h G V. Since u 2 — Wi + v 2 — V\ G V, we get by letting in the first of these inequalities h = u2—u\-\-v2—v\ and in the second h = u\ — u2 + v\ — v 2
and
242
Preiss: Almost differentiablity of convex functions
Adding these two inequalities and using that the Lipschitz constant of / on 2?(xo,3ro) is at most a, we infer that 2rj\\u2 - wi + v2 - vt\\ < 2a||*i - x2\\. Since u2 — ux and v2 — Vi belong to the spaces W and V f) Ker(?r), respectively, ||u2 - ui|| < 6 ||u2 - ui + v2 - vi\\. Hence IK - w i | |
-x2\\/r).
It follows that the set N C\ (w + W) is the image of the set D = {x £ X] there are u £ (w + W) D N and v £ V 0 Ker(Tr) such that x + u + v £ M fl B(xo,ro)} under the map
if and only if there is v £ V f) Ker(?r) such that u + v + x£MC\ B(x0, r0) and that tp is Lipschitz. Since D C X and dim(X) < dim(V^), we conclude that the dim(W) dimensional measure of N C\ (w + W) is zero, which clearly shows that N C\ (w + W) has empty interior in w + W. 1.4. Corollary. Let f be a convex continuous function defined on an open convex subset G of a separable real Banach space E and x £ G. Then
df(x) = f| ^-closure \{J{df(z)) z £ B(x9r) and dim(df(z)) < oo}] . r>0
Proof. The inclusion D follows from 1.1. To prove the opposite inclusion, suppose that x* £ df(x)y W is a finite dimensional subspace of E, e > 0 and r £ (0, dist(z, E\G)). It suffices tofindy £ B(x, r) and y* £ df(y) such that dim(df(y)) < oo and sup{|(/i,2/* - x*)\; heW, \\h\\ < 1} < e. Let TT denote the canonical projection of E onto the factor space E/W and let 5 £ (0, r) be such that for every w £ E/W with ||u; — TT(X)|| < 6 min{/(z) - f(x) - (z - x,x*) + e\\z - x\\; z £ S(a:, r) f| T T " 1 ^ ) } < er. Using Theorem 1.3 we find a point w £ J5/W such that ||u; — 7r(a;)|| < 6 and dim(df(z)) < dim(W) for every z £ (7 0 TT~1(W). Since VK is finite dimensional, there is y £ B(x,r) D ir"l(w) at which the function z £ B(x, r) fl T T - 1 ^ )
H->
f(z) - f(x) -(z-x,
x*) + e\\z - x\\
Preiss: Almost differentiablity of convex functions
243
attains its minimum. Moreover, our choice of 6 implies that y belongs to B°(x,r). Hence y + h G B(x,r) 0 ^(w) if h G W has sufficiently small norm and so for such h
Hy + h)-f(y)>(h,x*)-e[\\y-x
+
h\\-\\y-x\\]>(h,x*)-e\\h\\.
Thus the Hahn-Banach theorem provides us with y* G df(y) such that (h,y*) > (h,x*) — £||^|| whenever h G W has sufficiently small norm. Consequently, sup{|(/l,y*-a:*)|;*€ W, ||*|| < 1 } < £ . Since dim(9/(y)) < dim(VK) < oo, this proves our statement. 1.5. Corollary. Let E be a separable real Banach space E and let Q denote the set of all x* G E* for which there are x G E, rj > 0, and a finite codimensional closed linear subspace H of Ker(x*) such that (i) ||x|| = | | x l = (x,x*) = l (ii) lim^odl* + th\\ + \\x - th\\ - 2)/t = 0 for every
heH.
(iii) \\x + h\\ > 1 + rjdist(h,H) for every h G Ker(x*). Then Q is w*-dense in the unit sphere of E*. Proof. Since the set of norm attaining unit functional (which is the union of sub differentials at nonzero points) is it;*-dense in the unit sphere of i?*, 1.4 and the positive homogeneity of the norm imply that the union of sub differentials at those unit points x at which the dimension of the subdifferential of the norm is finite is also w*-dense in the unit sphere of E*. Whenever y* belongs to the subdifferential at such a point x and H = T(\\ • ||,x), then (i) and (ii) hold (with x* replaced by y*). Finally, using 1.2(ii), we approximate y* by another element x* of the subdifferential of the norm at x so that (iii) holds.
2. Determination of measures 2.1. Theorem. Whenever finite Borel measures fi and u over a separable Banach space E agree on all balls in E, then they agree.
244
Preiss: Almost differentiablity of convex functions
To prove this theorem, we use the following lemma from [2], 2.2. Lemma. Suppose that ft and v art finite Borel measures over a finite dimensional Banach space F, e* € F*, and D C {e € F; (e, e*) > 0} is a nonempty open cone such that D O Ker(e*) = {0} and such that ft and v coincide on every translate of D. Then the images of jl and v under e* coincide. Since measures on E are determined uniquely by their Fourier transform, it suffices to prove that the images of \i and v under all functionals belonging to a u;*-dense subset Q of the unit sphere of E* coincide. We prove that this holds with the set Q defined in 1.5. Let x* £ Q, x G E, and let H be a, subspace of Ker(z*) such that the properties 1.5(i)-(iii) hold. Then the set C := U£Li B°(nxyn) is an open convex cone in E. Moreover, since the union is nondecreasing, p. and v coincide on every translate of C. Let K be the canonical projection of E onto the factor space F := E/H and let D := K[C]. Using the fact that H C Ker(x*), we write x* = e* o K where e* 6 F*. Since 1.5(ii) implies that C + H = C, the image measures fi := /c[/x] and v := K[U] coincide on every translate of D. Finally, 1.5(iii) implies that D C {e € F\ (e,e*) > 0} and D H Ker(e*) = {0}. Hence Lemma 2.2 implies that the images of jl and v under e* coincide, which, because of x* = e* o /c, shows that the images of // and v under x* coincide.
References [1] R. R. Phelps, Convex Functions, Monotone Operators, and Differentiability, Lecture Notes in Mathematics 1364, Springer-Verlag, Berlin • Heidelberg • New York, 1989. [2] D. Preiss and J. Tiser, Measures on Banach spaces are determined by their values on balls. (To appear.) [3] L. Zajicek, On the differentiation of convex functions in finite and infinite dimensional spaces. Czechoslovak Math. J. 29 (104) (1979), 340-348.
WHEN E AND E[E] ARE ISOMORPHIC C.J. Read
Cambridge University, England
ABSTRACT
If E is a Banach space with 1-symmetric basic ( e ; ) ^ , the Banach space E[E] has an unconditional basis (e,j)?3=1 with norm || £*tj c tjll = II S c «ll E*«i c ilU WEi,j
i
3
If E is c0 or £p{\ < p < oo) it is easy to see that E[E] is isomorphic (indeed isometric) to E itself; however, this is not true in general, and at the recent conference in Austria, A.V. Bukvalov asked whether we could have E[E] isomorphic to E as a Banach space, for any other space E with symmetric basis. Here we show that this can be done, by methods which are strongly reminiscent of the present author's first ever paper [3]. For completeness we give a sketch proof of a related fact, namely that if E[E] is isomorphic to E as a. Banach lattice, then E must be either c0 or £p. I understand that isomorphisms between E and E[E] are helpful in the theory of Sobolev spaces, and this was the reason for the question being raised.
§1. Introduction.
We shall use the following notation. E will denote a Banach space with 1-symmetric basis (e«)£i» o n th e ( n o t closed) linear span lin {e;} we may define the £p norms (/ < p < oo) such that
1=1
\i=l
N
a n d similarly t h e c 0 - n o r m || ]£) Ajei||c 0 = m a x i |Aj|. t=i
Usually the norm on E will satisfy ||.||Co < ||.|| < ||.||^. Not unnaturally, we shall write £p (respectively c0) for the completion of (lin {e^}, ||.|| p ) (respectively, (lin { e j , ||.||c0))- T n e
246
Read: When E and E[E] are isomorphic
space E[E] is the space of all double series x = ^ ^ijeij such that
II £ *ll £ *
X
^\\E\\E
< oo.
(1.1.1.)
3
Note that E[E] and E[E] 0 E are isomorphic (even as Banach lattices, given their natural lattice structures). E[E] is just a countable number of copies of E summed in a certain symmetric way - adding on one more copy makes no odds up to isomorphism.
§1.2 Averaging projections oo
Let E be a Banach space with 1-symmetric basis ( e ; ) ^ ; let N = |J O{ be a partition of the natural numbers into countably many disjoint finite subsets
\i=i
where
(1.2.1)
where = denotes Banach space isomorphism. This result will be found in [1], p.117, and it is the one "external quote" necessary for a complete understanding of this paper. Now if E has a 1-symmetric (or even just a 1-unconditional) basis, then E[E] also has a 1-uncoditional basis (e«j)*j=i' ^ *s o u r tas ^> m ^ e n e x t section, to give a sketch proof
Read: When E and E[E\ are isomarphic
247
of the result alluded to in the Abstract, namely that if E and E[E] are isomorphic as Banach lattices, then E must be either £p or c0.
§2. Lattice isomorphisms. Now if E[E] is isomorphic to E as a Banach lattice, then the isomorphism must send "atoms to atoms", i.e., where ¥:n->(#n),^(n)) in some bijection N —• N x N. Because E has a symmetric basis, it doesn't matter which bijection we choose. So any bijection \P will do; now take any normalised block basis (w*)£i °f E, and for all n we can pick a # such that the vectors u\ ...un are mapped onto n distinct "rows" of E[E], and hence the (finite length) basic sequence u\ ... un is uniformly equivalent to the subsequence e\ ... en of the standard basis of E. So the whole sequence (u;) is uniformly equivalent to (e;), and E has "perfectly homogeneous" basis (all block bases are uniformly equivalent to it). Therefore by Zippin [5], E is CQ or £p. We now prove the main result of this paper, that E can be isomorphic to E[E] as a Banach space, without E being isomorphic to c0 or £p.
§3. Proof of the main result. We wish to construct a "symmetric norm" ||.|| on a vector space V with basis (c,-)^, such that the completion E of this normed space will satisfy E = E[E] as a Banach space, though the isomorphism will not be a lattice isomorphism. Let us begin by defining some very simple symmetric norms on V. Defintion 3.1 For each n G N let ||.||
HEW-^f sup f ^ A ^ ;
(3.1.1)
where 5(N) is the group of permutations on N. This norm just takes the sum of the n largest absolute values of the coefficients Aj, and then scales the answer by a factor (log n)/n, so
248
Read: When E and E[E] are isomorphic
[This scaling is done for ease in constructing a symmetric norm which obviously isn't equivalent to the CQ norm, but increases more slowly than any £p norm] Note that
We shall not be using all of the ||.||( n )'s in our construction, so for the sake of being definite, let us write and define
l|.||; = ll-ll ( n i ) -
Then define a map J : l\ —• t\ by
J(a) = (||a||0£1 So J (a) is a positive sequence in t\ consisting of the sequence of norms We know that hence
say. Recall that we have chosen ourselves a bijection $ : N -> N x N . We now use this map to define some more symmetric norms, in the following way. For x 6 V let
N l * = IMIc where ^ is the linear map sending e; —• e<£(i),^(
Then an elementary piece of induction tells us that \\x\\Ei is an increasing function of i\ furthermore it is a bounded function of i, since if for all x we have \*\\Et < P.\\*\W
Read: When E and E[E] are isomorphic
249
then WXUIB,] < P'-IMIMM
therefore MEi+l=\\x\\C0+\^Jx\\Ei[Ei] <\\x\\tl +P2\\Jx\\tl
Hence, for aU i, \\x\\El < 2\\x\\tl. So we may define \\x\\E = lim; \\x\\Ei and ||x|| £ = ||*||co + ||« Jx\\E[E]
(3.2)
OO
Now let N = \J<Ji be a partition of the natural numbers into sets cr,- of size <7j = raj. Let 1
n be the averaging projection on E associated with the partition (CTJ)I°. Then the image of 7T is the closed linear span of the vectors xi = Yl ej- ^ e define a unique linear map $ : lin{xj} —> E[E] by requiring that
This brings us to our first main theorem: Theorem 3: $ is continuous and extends to an isomorphism of Iran with Proof: It is clear (because $ : N - > N x N i s a bijection) that $ has dense range in We must therefore establish that both $ and $~ 1 are bounded linear maps where defined. To begin with, we note by (3.1.1) that ||:r;||(n*) = logh;. N
So if x = Y^, ^i Xi 6 linjzi}, since the Xi are disjointly supported and ||.||^n^ is a symmetric i=i
norm, we have N
J0r)>]T log n^Ail-e; where ">" is in the sense of the lattice ordering on i\. So, N
250
Read: When E and E[E] are isomorphic
and so N 1=1
N
(for E[E] has a natural unconditional structure)
= 11*0011
(i)
However, by (1.2) we have \\X\\E = \\X\\C0 + \\*JX\\E[E]
(2)
so (1) and (2) tell us that ||z|| > ||$(s)||,*.e.$ is a contraction. Getting a bound on $-* is not quite so simple, but is broadly similar. The idea is that if x = ]T Aj#j = Ylaiei s a v ^ e n J(x) i s roughly equal to To implement our idea, write |||#||| = maxi{|A;|.lognj} = ||^(^) Then, |||x||| < ||^(aj)||. Now we know that
however, the coefficient of ej in J(x) is
}&
3=1
i=l
"'
< \ogni.\Xi\ + 111*111.2"' (say). Hence, J(») < (Slo«»>i.|Ai|.eO + |||*|||
}='+!
Read: When E and E[E] are isomorphic
251
(since | | . | U < 2 . | | . | k ) ,
Therefore, \\*\\E=\\X\\C
<\\x\\c
<6||*(*)||. Thus, H^"1!! < 6. This is proves Theorem 3. But Theorem 3 gives us our result. For (1.2.1) tells us that E = E © Irrnr (since IT is an averaging projection), and Theorem 3 adjusts this to E = E © E[E]. But we have already made the trivial observation that E[E] £ E[E] 0 E. Therefore E £ E[E], (incidentally it's then obvious that E and £[£[£]] are isomorphic, and so on). Furhermore we do not have E isomorphic to CQ or to any ^p, because for selected values of n we have
which of course doesn't happen in CQ or £p. Thus the question of Bukvalov is answered, and we find that we can have E and E[E] isomorphic as Banach spaces without E = £p or co. Note also that, by our earlier arguments, this is an example of Banach space isomorphism without Banach lattice isomorphism.
Note: The referee pointed out that, as well as c0 and Hv, the universal space U\ of Pelcynski also has the property that U\ = £7i[{7i]. For it is known to have a symmetric basis, and J7i[f7i] obviously contains a complemented U\ so it is another universal space with unconditional basis, therefore it is U\. For a discussion of U\ see Pelcynski [2] or Singer [4], pp. 547-550.
I would like to thank Walter Schachermayer for organising a jolly good conference in Strobl, which I found very stimulating - this is my claim, at any rate, and here at least is a paper in support of it! But he knows, and we all know, that we had a very good time.
252
Read: When E and E[E] are isomorphic
4. References [1] J. Lindenstrauss and L. Tzafiri, Classical Banach Spaces, Vol. 1, Springer, (1977). [2] A. Pelcynski, Universal bases, Studia Math. 32, 247-268, (1969). [3] C.J. Read, A Banach space with, up to equivalence, precisely two symmetric bases, Israel J. Math., Vol. 40, No. 1, (1981). [4] I. Singer, Bases in Banach spaces I, Springer, (1970). [5] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4, 265-272, (1966).
A note on Gaussian measure of translates of balls Michel Talagrand(*)
Abstract. Consider a centered Gaussian measure pi on a separable Banach space X. W. Linde [2] proved that if B is the unit ball of X, the function x —• fM(B) — /i(x+J5) is Gateaux-differentiable at zero, and asked whether n(B) - n(x+B) < C\\x\\2. We show that this is not the case; actually, if X = £p, 1 < p < 2 , and /x is "diagonal", and not supported by a finite dimensional space, then no estimate /x(-B) — fj,(B + x) = o(||x||p) can hold.
Denote by (e n ) n >i the canonical basis of £p, and by B its unit ball. Throughout the paper, we assume 1 < p < 2, and we denote by (gn) an i.i.d. sequence of JV(0,1) random variables. Consider a sequence an > 0, such that £ a£ < oo, and the law /i on £p of £ n>l
Ongnen.
n>l
Proposition. For some constant c depending on /x and p only, we have for all k (1)
/i(£) - /x(B + akek) >capk = c\\akek\\r. Condition (1) implies that we cannot have an estimate (JL(B)—H(B+X) = o(||a;||p). In particular,
for p = 1, the map x —• fi(B) — fj,(B + x) is not Frechet differentiable at zero. This complements the results of [3], where it is shown that
Proof. We fix k, and we denote by nk the law of £ angnen. We set, for v > 0 hk(v) = (*) Work partially supported by an N.S.F. grant.
254
Talagrand: Gaussian measure of translates of balls
Essential to our approach is the fact, consequence of a well known result of C. Borell [1], that log hk is concave. For 0 < t < 1, we set
so that f'(t)
(2)
By Fubini's theorem, we have (3)
fi(uek + B) = — \ = f f(t aky/2ir Ju-l
We set
c""2/2 dv v == — U / • £
Integrating (3) by parts, we get
so that
Since / ( - t ) = /(t) and O(x) - O(2r) = - O ( - x ) + O(-2f), we get (4)
fi(B) - ti{uek + B) = f
-f'(tmt/ak,
u/ak) dt
Jo
where ¥(*, u) = 2O(t) -
A*CB) — n(B — akek)
= aPk
Talagrand: Gaussian measure of translates of balls
255
where 6(t) = tP" lv F(t, 1) > 0 . Since log hk is concave, for v > 1 we have h'k(u)>loghk(v)-\oghk(u) hk(u) ~ v—u so that, for 1 > u > 1/2,
> tik(u) tik(u) >
(6)
v -t tii
log-- ^^ >-h>-h hhkk (( -- )) log k[-) log - ^ k[-) V 22// hk(u) v \2/
Obviously for all k and t, hk(t) = fik(tB) > fi(tB). There is no loss of generality to assume that the sequence (an) decreases. Since f*(B) is clearly a decreasing function of the sequence (a n ) (for the componentwise order) we have nk(B) < /*I(JB) . If we choose v large enough that p(yB) > m(B), we get from (6) that for 0 < u < 1/2,
Together with (5), this implies the result. D Acknowledgement. The problem investigated in this paper is motivated by questions of W. Linde. References [1]
C. Borell. The Brunn-Minkowski in Gauss Space, Invent. Math. 30, 207-216,1975.
[2] W. Linde, Gaussian measure of translated balls in a Banach space, to appear in Teor. Veroj. i. Primenen. [3]
M. Ryznar and T. Zak, "The measure of a translated ball in uniformly convex space", manuscript, 1989.
[4]
T. Zak, On the difference of Gaussian measure of two balls in Hilbert spaces, Probability Theory on Vector Spaces IV, Lecture Notes in Math 1391, p. 401-405, Springer Verlag, 1989.
256
Talagrand: Gaussian measure of translates of balls
Equipe d'Analyse - Tour 46
Department of Mathematics
U.R.A. au C.N.R.S N ° 754
The Ohio State University
Universite Paris VI
231 West 18th Avenue
4 PL Jussieu
Columbus, Ohio 43210-1174
75230 Paris cedex 05
Sublattices of M(X) isometric to M[0,1] Lutz W. Weis Louisiana State University Baton Rouge, Louisiana 70803
1. Introduction Let X,Y always denote infinite Polish spaces and M(X) stands for the Banach space of Radon measures on X with the variation norm. While the isometric type of sublattices of Li(X,p) for some \x £ M(X) is completely understood (they are either /i, Li[0,1], or 11 © Li[0,1], see e.g. [S] III Prop. 11.2) the structure of sublattices of M(X) is much more complicated. Interesting examples of such sublattices are the Henkin-measures on the unit sphere of C n for n > 1 (see [Rn], Chap. DC), Rajchman measures on the unit circle (see [Ke], Chap. IX), invariant mesures of a family of measurable tranformations of X (see [Ph], Chap. X) and, more generally, the invariant measures of a If-sufficient statistic in the sense of Dynkin (see [Dy], [Ma]). The first two examples are actually bands in M(X) and there is a very nice characterization of such bands in terms of the compact subsets of X that they annihilate due to Mokobodski ([Ke], Chap. IX. 1). The last two examples are usually true sublattices of M(X) isometric to M(0,1). In this note we characterize sublattices L of the latter kind, (i.e. L is isometric to M(0, l))in terms of the existence of strongly affine projections, the iy*-Radon-Nikodymproperty, martingale compactness, a choquet-type integral representation theorem and finally in terms of the embedding of their unit sphere into M(X) (see section 2 for precise statements). These characterizations are derived from general integral representation theorems of Bourgin-Edgar ([Bo], [Ro]) and results on orthogonal kernels by Mauldin, Preiss and v. Weizsacker ([Ma]), but in some cases we can give (in our lattice setting) more direct and
258
Weis: Sublattices of M(X) isometric to M[0,1]
easier proofs than the ones known in the literature. Let us recall some definitions. Let F be a separable Banach space and U a bounded convex subset of F*. U has the w*-RNP if for every probability space (H,^, P) and bounded linear operator T : Li(H,P) —> F* with Tf eU for all / in the positive sphere of Li(P) there is a immeasurable function
U
=w
* ~ Jl
with
Tf = w* - J fdP
It is well known that every tu*-closed U has the w*-RNP and for a line-closed, tu*-Borel u;*-measure convex U the w*-RNP is equivalent to the iu*-martingale compactness, i.e. every {/-valued martingale (fn) with fn finite-valued for each n, it;*-converges almost every where to a ^/-valued function / . (see e.g. [Gh], Chap. I). Recall that U is w*-measure convex if the u/*-barycenter of every w*-Radon measure supported on U belongs to U. A function t : U —• F* is strongly affine if it is universally measurable for the w*-topology and for every probability measure P on U with barycenter x G U we have that t(x) is the barycenter of the image measure Pot" 1 on t{U). (see [D-M] Chap. X, Sect. 3, No. 50). The following notion arises in mathematical statistics (see [Dy]). Let (X, B) be a measure space and U a family of measures on (X, B). A sub-a-algebra E of A is called H-sufficient for U if there is a stochastic kernel
(MZ)XGX
with \ix G U such that for all B-measurable
/ : X -> IR we have for all fi G U
-I i.e. g(x) = J fdfix is a common conditional disbribution of / given E for all fx G U. Throughout we refer to the cr(M(X), C&(X))-topology on M(X) as the u;*-topology which on the positive unit ball of M(X) is induced by the a(F*,F)-topology, where F is the separable Banach space of functions on X uniformly continuous with respect to a totally bounded metric on X.
Weis: Sublattices of M(X) isometric to M[0,l]
259
2. The Result L denotes a fixed, norm-closed, non separable sublattice of M(X), the space of Radonmeasures on a Polish space X. P(X) is the set of probability measures on X and r/ = {A6L:||A|| = l, E — {A G U :
A>0}
A an atome of L}.
Notice that E is the set of extreme points of U. M(fi) = {y G M(X) : a is /z-absolutey continuous} Theorem.
Let L, U, E as above and assume that U is iy*-measure convex and w*-
analytic and E is tu*-Borel. Then the following four conditions are equivalent: a) There is a strongly affine order isometry J of M(0, l) onto L. b) For every fi G U, there is a unique ^-probability measure on E with \i as its w*barycenter. c) U has the iu*-Radon-Nikodym property (or, equivalently, is martingale-compact.) d) For every fi G L+ there is a immeasurable kernel x G X — • fix G E such that the positive projection P : M(/z) —> M(IL) fl L is given by P{\) = w* - J fixd\
for
AG
For the following conditions e), f), g) we always have e) <& f) => g) =>• a) and if we assume Martin's axiom all of the conditions are equivalent. e) There is a positive, contractive projection P of M(X) onto L, which is also strongly affine. f) There is a countably generated a-algebra E of universally measurable subsets of X such that E is if-sufficient for U. g) There are iu"-universally measurable, w*measure convex sets Mn so that
P(X)\U = [JMn.
260
Weis: Sublattices of M(X) isometric to M[0,1]
Remarks: 1) The implication c) => b) follows from Edgar's choquet-type theorem for simplices with tu*-RNP ([Bo]) and its extentions (see e.g. [Ro]). We take advantage of our more specialized lattice setting to give a different and simpler proof below. 2) Of course there is always a positive projection P : M(/JL) — • M(fi) D L which — as a conditional expectation operator — is given by a kernel of measures (fix) on X. So the essential requirement in d) is that /xy can be chosen in U. Similarly, there is always a positive projection P : M(X)
— • L ([S] III, Prop. 1.1.). Therefore the main point in e)
is that P is in addition strongly affine. A typical projection without the latter property is the projection P : M[X)
—> M(fx) given by Lebesgue's decomposition. Notice that the
unit ball of M(fz) has no extreme points. 3) The set theoretic assumptions enter through the separation theorems of Mauldin, Preiss and v. Weizacker in [Ma]. They only use the weaker (but less well known) assumption of "existence of medial limits" (see e.g. [D-M], Chap. X, Sect. 3, No. 55). 4) The condition ufix G U" in the definition of if-sufficiency, is essential; if we replace 'insufficiency' in g) by the usual notion of 'sufficiency', then condition f) would be satisfied for L = M ( X , M ) and E the Borel sets in X. Also, the counterexamples in [Ma] show, that in general we cannot choose E as a a-algebra of Borel subsets of X. 5) While d) =$> e) may very well depend on set theoretic assumptions we suspect that a) =>• g) holds in general. This is certainly true if U is a w*-G$ set since then we can apply the results of Ghoussoub and Maurey on tt;*-i?s-embeddings ([Gh], Chap. I). Condition g) is of course inspired by their "f^-embeddings". 6) The assumptions that U is iu"-measure convex and E is tu*-Borel cannot be dropped completely. Of course a) implies both of these properties but some of the other conditions do not. E.g., if Y is an uncountable, universal zero set in X, then the space L of Radon measures concentrated on Y satisfies b) to d) but not a) although E is universally measur-
Weis: Sublattices of M(X) isometric to M[0,1]
261
able and U is measure convex. Also, the space L of purely atomic measures on X satisfies b) to d) but U is not measure convex. 7) The requirement in a) that J is strongly affine cannot be dropped. Just consider an isometry that maps the atomic measures on [0,1] onto the atomic measures on [0, -|] and the diffuse measures on [0,1] onto the diffuse measures on [•§, 1]. Corollary: A band L C M(X) satisfies the assumptions of the theorem and one of the conditions a) to b) if and only if L is of the form L = M(Y) for a Borel subset Y C X. Proof:
Since L is a band, E must consist of point measure. Now choose Y C X such
that E = {6X : x G Y}.
I
The following classical exampe illustrates the conditions of the theorem. Example: X is a Polish space and B the Borel sets of X. Let 4>n : X —> X be a sequence of Borel measurable mappings and L be the set of measures which are <£n-invariant for all n; i.e. L = {fie M(X) : /x^-^A)) = fi(A)
for alL4 £ B, n G IN}
There are elementary arguments that show that L is a sublattice of M(X) and the extreme points E of U are precisely the ergodic measures ([Ph], Sect. 10). Also, U is tu*-measure convex and
P(X)\U = (J M±.A
e
P(X) : n{UAj)) > ^Aj) +
where Aj is an enumeration of a countable subalgebra Bo of B which separates the points of X. It is easily checked that all M^^s
are iu*-Borel subsets of P(X) and tu*-measure
convex. Now g) => b) of the theorem gives the usual representation of invariant measures
262
Weis: Sublattices of M(X) isometric to M[0,1]
as integrals of ergodic measures ([Dy] Sect. 6, [Ph]). The classical proofs show directly that the a-algebra E = {A G B : n{(j>~lAAA) = 0 for all /x G L and n G IN} of {<£n}-invariant sets is sufficient as in condition f) or, if n = ^ n , that
has a weak limit which defines a projection of M(X) onto L as in e). 3. Proofs a) => c) Let T : Li(0,P) —> L be an operator that maps the unit sphere of Li(H,P) into U. By [Du], VI.8.6, there is iu"-measurable function
- J
/GLi(P).
Since J is strongly affine we get
T(f)=w*-J(Jod>)fdP. c) => d) We may identify M(fi) with Li(X,^) and M(//) n L with Li(X,E,^) where S is a sub-
Pf(x) = I f{y)dnx(y)
for
and x
G X.
Weis: Sublattices of M(X) isometric to M[0,1]
263
See e.g. [D-M] Chap. Ill, No. 70-72. One can check that this implies PA = «;*- J »xdX{x)
for A G M(M) .
Indeed, using the standard properties of conditional expectations we get for / G Cb(X) and A G M(M) with dX = gdfi, g G Li(/x)
PX(f) = J P(g) .fdv=f
E{g\Z)fdp
- E(f\E)dp =JJ
fdnxg(x)dfi
It remains to show that fix G E for /z-almost all x G X. First of all, we can assume that \ix G U. Indeed, since U has the tu*-RNP there is a kernel firx G U for all x G X such that for the operator P : M(fj) —> M(X) we have PA = w*- I firxdX(x)
for
A G
Since this kernel representation is unique up to a /x-nullset we have fix = fj,'x G U
fM-di.e. Now assume that there is a Borel set A C {x G X :
Hx&E}
with positive ^-measure. A — f(\ J\
r\
C-Ts/A*f}
— 1 l / \ j J / l vZ - " y^ •**• • ^^ _^ /» _^ r"Xj
is an analytic subset of the Polish space UM(X)
X
f} -A \
\ r£ ii X
*^ /
^* /
^*>
H"Xj
-X" an< i by applying the von Neu-
mann selection theorem ([D-M], Chap. Ill, No. 81) to the coordinate projection UL x A —• A restricted to A we find a it;*-universally measurable kernel x G A —> Ax G U with Ax ^ O, Xx ^ fix and O < Az < fix. By Lusin's theorem
264
Weis: Sublattices of M(X) isometric to M[0,1] there is an AQ C A with fi(Ao) > O such that x G AQ —• Ax is w * -measurable and we define
(
AX(X)~1AX for x G Ao fix
for x G X\A0
the immeasurable kernel x G X —• fix G U still satisfies }irx((p~1((p(x))j = 1 for /i-almost all x G X. This implies again that P = i£(-|E) on M(/z), or PA = J fJ.'xd\(x)
for A G Af (A«).
The uniqueness of this kernel representation gives the contradiction that JJ,X = \LX for ^-almost all x G X. d) => b) Given fj, G U, we choose a kernel x G X —• /xx G .E as in d). As in c) =» d) we see that for a Borel function / on X
-I where E is the sub-a-algebra generated by the map <j> : X —> JE, X —> /^x. Now, if Q = /x o 0" 1 , then for every /x^ = ^(A) -1 /i(A n •), A G E, we have
For A = X we see that \i is the barycenter of Q on 1£. As a preparation for the proof of uniqueness, we choose an increasing sequence of a-algebras each generated by a finite set B n of atoms such that (J
f o r a11 B e
In particular, the ranges of
B
n-
Weis: Sublattices of Af(X) isometric to A/[0,l]
265
(where terms with Q(B) = 0 are omitted) are precisely the atoms of L\ (o{A n ) , A*) • Let Qn — QoF~l be the distribution of this (>ln,Q)-martingale on U. Now let R be another probability measure on E with barycenter /x. We form the martingale
Gn{u)= E with respect to {An,R) and denote by Rn = RoGnx its distribution on U. We are going to show that in the Choquet order of measures on U Rn is "almost" smaller than Qmn for an appropriate n m . Indeed, since b(R) = \i and U is measure-convex, Gn has its values in M(fi) H L = Li(S,/z) and for every n we can find an m = m(n) such that for vB = E(R(B)~1b(R\B)\Am(n)) WRiB^biRl^
we
have
- t/B\\ < - for all B G Bn n
and
If vB =
E
«AMA we get
and therefore Q{A) =
E
i2(5)a^ and
£
A
a^ = 1.
Put Rn =
6UB. In order to show that Rn < Qm(n) in the Choquet order, we choose a continuous convex %/> : U —> IR. Then
BGBn
= (n)
J U
For the martingale Fn with respect to (A n , Q) we have FnM -* ^ in the topology of U for Q-almost all v G -E. Therefore, Q n —• Q weakly on U. Similarly, since
266
Weis: Sublattices of M(X) isometric to A/[0,l] ^n(^) —• if -R-a.e. on U implies that Rn —> R and Rn -± R weakly on U. Since Rn < Qn(m)
we
g e t for n —• oo that R < Q in the Choquet order of U.
But since J? and Q are supported by the extreme points E of U, it follows that # = Q (see e.g. [Ed], theorem 2.2). b) =>• a) Since L is non-separable, b) implies that E is a uncountable Borel subset of the Polish space P{X). Then there is a Borel isomorphism j : [0,1] —• E ([D-M], Chap. Ill, No. 80). Define J : M+[0,l] -> L+ by taking J(A) as the w*barycenter of the measure Aoj" 1 on E. By b) this J is bijective and standard lattice arguments show that A _L v implies J\ _L Jv for A, v £ M+[0,1], Now J(A) = J(A+) - J(A~) extends J to a lattice isometry of M[0, l] onto L. J : M[0,1]) -> M(X) is u>*-Borel measurable and strongly affine. Indeed, for a probability measure on P([0,1]) with barycenter fi £ P[0, l] and a Borel function / on X and / : P(X) - IR, /(i/) - / /di/, we get
Po J-^/) = J(f)oJdP = J
=f
= Mf)
i.e. J(fi) is the barycenter of PoJ~1. a) => e) Assuming a), y G [0,1] —• jxy := J($y) is an orthogonality preserving kernel in the sense of [Ma]. For such a kernel it is shown in [Ma], theorem 4.3, that — assuming the existence of medial limits — there is a universally measurable tp:X —• [0,1] such that fiy^P~l{y)) = 1 for all y G [0,1]. Define fix = £
PA = J fixd\{x)
for
AGM(X).
Since U is measure convex we have P(P(X)) C U and it is easily checked that
Weis: Sublattices of M(X) isometric to M[0,1]
267
PA = A for X £ L. The same argument that showed in b) => a) that J is strongly affine also shows that P is strongly affine, i.e. P is the required projection. e) => f) Given P we consider the universally measurable stochastic kernel ftx = P6X G U, x £ X. For A G P(X) think of A as the induced measure o n l = {6X : x G X} with barycenter A. Since P is strongly affine we get that . - . . - / ,
Let E be the smallest sub-a-algebra of the universally measureable sets of X that makes the map x G X —• /zz G P{X) measurable. For a fixed fj, G U we again identify M(fi) with Li(X,fi) and Li(X,ft) D L with Li(Eo,/x) for an appropriate sub-<7-algebra Eo of Borel sets of X. Then P induces on
LI(EQ,AO
the conditional expectation operator 2£(-|E). Observe that for all
/ G L\(X\ii) and bounded Borel function g on X :
= P(g • M )(/) -Hence £(/|E 0 ) = / fdfix
J\xJ\{x{f)g{x)dn{x)
/x-a.e.
Since functions of the form x —* fix(f), f G C&(X) generate the a-algebra E and each of them equals /j-a.e. the Eo-measurable function ^(/|Eo) it follows that modulo ^-zero sets EQ and E are identical and that E is a if-sufficient <7-algebra for U with statistic (fix)x^xf) => e) By f) there is a sub-cr-algebra E of universally measurable sets of X and a universally measurable stochastic kernel (fix)xex with fix G U such that for all Borel functions / on X and all fi G U we have
268
Weis: Sublattices of M(X) isometric to M[0,l]
Define PA = w*-f fiydX for A G M(X).
Since U is measure convex, the image
of P must be contained in L. By (++) (and the same arguments used in e) =>• f)) P is a projection on every M(/x) for al /x G U. Hence P is a strongly affine projection onto L. e) => g) A measure /i G M+(X),
\\fi\\ < 1, belongs to U if and only if P\i = \i or
Pfj,(Bn) = fi(Bn) for all n, where {Bn} is a countable subalgebra of the Borel sets that generates all Borel sets. Therefore
where
y = { A G P(X) : PX(Bi) < X(B{) - i J
% = {A G P(X) : P\(B{) > X(B{) + i Since P is a it;*-universally measurable kernel operator it follows that Mty and JV»y are all w*-universally measurable and iu*-measure-convex. Indeed, if Q is a probability measure on Mty with barycenter jx, may use that P is strongly affine and obtain
Pn(Bi) = / PX{Bi)dQ{X) < j X(B{)dQ{X) - i = p(B{) - * M
Mij
ij
g) =>- a) uses an idea of Edgar and Wheeler (see also [Gh] Theorem 1.3). Let T : Li(O,P) —• M(X) be a bounded linear operator which maps the unit sphere of Za(P) into U. Since P(X) has the it;*-RNP there is a w"-measurable
: n —> P{X) with Tf = w*-J
for / G Li(P)
We want to show that
P(
Weis: Sublattices of M(X) isometric to M[0,l]
269
Otherwise, there is a Mty (or Nij) so that A =
Tf = w*- I (t)f(t)dP(t) = P(A)-1 I cf>(t)dP(t) j
= I
J A
udQ{u) e Mij
where (J = (/oP)o~ x is the distribution of
|
270
Wcis: Sublattices of M(X) isometric to M[0,l]
References 1) [Bo] R. D. Bourgin and G. E. Edgar : Non-compact simplices in Banach spaces with the Randon Nikodym Property, J. Funct. anal. 23 (1976), 162-176. 2) [D-M] C. Dellacherie and P. Meyer : Probabilites et potentiel, Herman, Paris. 3) [Du] N. Dunford and J. Schwartz : Linear Operators I, Interscience Publishers, New York (1958). 4) [Dy] E. B. Dynkin : Sufficient statistics and extreme points, Ann. of Prob. 6, 1978, 705-730. 5) [Ed] G. A. Edgar : On the Radon-Nikodym Property and Martingale Convergence, in : Vector Spaces Measures and Applications II, Lecture Notes in Mathematics 654, p. 62-76, Springer Verlag, Berlin-Heidelberg-New York, 1978. 6) [Gh] N. Ghoussoub and B. Maurey : i/$-embeddings in Hilbert space and optimization on G6-sets. Memoirs of the AMS 349, Providence, 1986. 7) [Ke] A. S. Kechris and A. Louveau : Discriptive Set theory and the Structure of Sets of Uniqueness, Cambridge University Press, 1987. 8) [Ma] R.D. Mauldin, D. Preiss and H. v. Weizsacker : Orthogonal Transition kernels, Ann. of Prob. 11, 1983, 970-988. 9)
[Ph] R. P. Phelps : Lectures on Choquet's theorem, D. von Nostrand Comp., Princeton-New York, 1966.
10) [Ro] H. Rosenthal : Sub-simplices of convex sets and some characterizations of simplices with the RNP, in :Banach Space theory, Comtemporary Mathematics 85 (1987), 447-465. 11) [Ru] W. Rudin : Function theory in the unit ball of C n , Springer Verlag, New YorkHeidelbert-Berlin, 1980. 12) [S] H. H. Schaefer : Banach lattices and Positive Operators, Springer Verlag, BerlinHeidelberg-New York, 1974.
