The Rademacher Cotype of Operators from l N

Transcription

1 The Rademacher Cotype of Operators from l N SJ Montgomery-Smith Department of Mathematics, University of Missouri, Columbia, MO 65 M Talagrand Department of Mathematics, The Ohio State University, 3 W 8th Avenue, Columbus, OH 43 Equipe d Analyse Tour 46, Université Paris VI, 4 Place Jussieu, 753 Paris Cedex 5 ABSTRACT: We show that for any operator T : l N Y, where Y is a Banach space, that its cotype constant, K ) T ), is related to its, )-summing norm, π, T ), by K ) T ) c log log N π, T ) Thus, we can show that there is an operator T : CK) Y that has cotype, but is not -summing AMS Classification: Primary 46B, Secondary 6G99 Introduction The notation we use in this paper is loosely based on that given in [L T], [L T] and [P] We let ε, ε, be independent Rademacher random variables, that is, Prε s = ) = Prε s = ) = A linear operator T : X Y is said to have Rademacher) cotype p p ) if there is a constant C < such that for all x, x,, x S in X we have S T x s ) p ) p S C IE ε s x s The smallest value of C is called the Rademacher) cotype p constant of T, and is denoted by K p) T ) These definitions extend to spaces in the obvious way; a space X has cotype p if its identity operator has cotype p

2 THE RADEMACHER COTYPE OF OPERATORS FROM l N We define the p, q)-summing norm of a linear operator T : X Y, denoted by π p,q T ), to be the least number C such that for all x, x,, x S in X we have S T x s ) p ) p S ) q C sup x, x s q, where the supremum is taken over all x in the unit ball of X We call a p, p)-summing operator a p-summing operator, and write π p T ) for π p,p T ) We say that the operator is p, q)-summing p-summing) if π p,q T ) < respectively π p T ) < ) If p <, and q, then we let L p,q µ) denote the Lorentz space on the measure µ We refer the reader to [H] or [L T] for details, but just note that the L p, norm may be calculated using f p, = µ f > t) p dt = p s p f s) ds, where f denotes the non-decreasing rearrangement of f The basic motivation behind this paper is in classifying operators from CK) that factor through a Hilbert space, where CK) denotes the continuous functions on the compact Hausdorff topological space, K The first result in this direction is due to Grothendieck, which states that any bounded linear operator CK) L factors through Hilbert space This was generalized by Maurey [Ma], allowing L to be replaced by any space of cotype, to give the following result see also [P]) Theorem Let T : CK) Y be a linear operator, where Y is any Banach space Then the following are equivalent: i) T is -summing; ii) T factors through Hilbert space; iii) T factors through a space of cotype However, we are still left with the following question: if the operator T : CK) Y has cotype, does it follow that it factors through Hilbert space? One way one might tackle this problem is to consider the, )-summing norms of such operators Jameson [J] showed that there is an operator T : l N Y such that π T ) c log N π, T ) Hence, if we can establish a strong relationship between the cotype constants and the, )-summing norms of such operators, then we can answer the above question in the negative To this end, we have the following the main result of this paper

3 MONTGOMERY-SMITH TALAGRAND Theorem There is a constant c such that for any operator T : l N Y, where Y is a Banach space, then the cotype constant is bounded according to the relation: K ) T ) c log log N π, T ) Corollary There is an operator T : CK) Y, where Y is a Banach space, that has cotype, but does not factor through Hilbert space Finally, before embarking on the proof of this result, we point out that for p >, the above problems have been completely answered Theorem 3 Let T : CK) Y be a bounded linear operator, where Y is a Banach space Then for all p >, the following are equivalent: i) T is p, )-summing; ii) T has Rademacher cotype p; iii) T factors through a space with Rademacher cotype p The implication i) ii) is due to Maurey [Ma] The third equivalence follows from the fact that any p, )-summing operator from CK) factors through L p, see [P] or Theorem 5 below), and that L p, has Rademacher cotype p, see [C]) Theorem 4 If p >, then there is a bounded linear operator CK) L p that is not p-summing We refer the reader to [K] 3

4 THE RADEMACHER COTYPE OF OPERATORS FROM l N Proof of the Main Result To prove Theorem, we need the following two results The first allows us to reduce questions about p, )-summing operators from CK) to the canonical embedding CK) L, K, µ) µ a probability measure), and is due to Pisier see [P]) Theorem 5 Let T : CK) Y be a p, )-summing operator, where Y is a Banach space, and p Then there is a Radon probability measure µ on K and a constant C p p πp, T ) such that for all x CK) we have T x C x Lp, K,µ) The second result is about Rademacher processes, and is due to the second named author for the proof, see [Ld T]) First we establish some more notation If T is a bounded subset of IR S, we write S rt ) = IE sup ε s ts) t T If B is a subset of IR S, we write N T, B) for the minimal number of translates of B required to cover D We write B S for the unit ball of l S, and B S for the unit ball of l S From now on, we take all logarithms to base Theorem 6 There is a constant c such that if T is a bounded subset of IR S, and ɛ >, then letting D = c rt ) B S + ɛ B S, we have rt ) c ɛ log N T, D) Now we will state the main result towards proving Theorem Proposition 7 There is a constant c such that if Ω, F, µ) is a probability space with N atoms, and x, x,, x S L µ) are such that S IE ε s x s, then x s L, µ) c log log N form Our first step in establishing this result is to restate Theorem 6 in a more suitable 4

5 MONTGOMERY-SMITH TALAGRAND Lemma 8 There is a constant c the same one as in Theorem 6) such that the following holds Suppose that Ω, F, µ) is a measure space, with Ω finite, and x, x,, x S L µ) with S IE ε s x s Then for all integers k, we may partition Ω into at most k measurable sets, find y, y,, y S, z, z,, z S L µ), and find x, x,, x S L Ω, F, µ) where F denotes the algebra generated by the partition), such that x s = x s + y s + z s, S S S ) IE ε s x s, y s c and z s c k Proof: Let T = see that there are k { xs ω) ) S : ω Ω }, and let ɛ = c k If we apply Theorem 6, we translates, t l + c B S + k B S ) l k ), that cover T We let the covering of Ω be the sets { A l = ω : x s ω) ) } S t l + c B S + k B S ), and if A l is non-empty, we choose ω l A l Define x s ω) = x s ω l ) if ω A l Now, if ω A l, we know that x s ω) x s ω) ) S c B S + k B S ), that is, there are y s ω) ) S c B S and z s ω) ) S c k B S, with x s ω) = x s ω) + y s ω) + z s ω) Lemma 9 There is a constant c 3 such that if Ω, F, µ) is a measure space with Ω finite, then the following hold i) If y L µ), then y, y y ii) If the smallest atom is of size a, then for all z L µ) we have z, c 3 + ) logµω)/a) z iii) If there are N atoms, then for all z L µ) we have z, N z Proof: i) We have that y, = y y µ y > t) dt dt = y y ) y 5 µ y > t) dt )

6 THE RADEMACHER COTYPE OF OPERATORS FROM l N ii) We have z, = z s) s ds a z + µω) a z s) s ds ) µω) z + ds µω) a s a c 3 + ) logµω)/a) z z s) ) ds ) iii) Let B, B,, B N be the atoms of Ω arranged so that z n), the value of z on B n, is in non-increasing order Also, let z N + ) = Then N n z, = µb m ) n= m= ) z n) z n + ) ) N n N µb m ) z n) z n + ) ) n= m= N N µb n ) z n) ) n= = N z as desired We remark that Lemma 9i) is a well known interpolation result, and is true for all measure spaces Lemma If Ω, F, µ) is a probability space with Ω finite, then y s, S y s Proof: This follows straight away from Lemma 9i) 6

7 MONTGOMERY-SMITH TALAGRAND Lemma is also well known and true for all probability spaces) In fact it is a reformulation of the statement that the canonical embedding CΩ) L, µ) has, )- summing norm equal to Lemma There is a constant c 4 such that, if Ω, F, µ) is a probability space with at most N atoms, then z s, S ) c 4 log N z s Proof: Let A Ω be the union of those atoms of measure less than N, so that µa) N By Lemma 9ii), we have that z s χ Ω\A, c 3 log N zs, and by Lemma 9iii), we have that z s χ A, N z s χ A Thus, we have that z s, zs χ Ω\A, + z s χ A, S ) c 3 log N z s + S ) N µa) z s S ) c 4 log N z s, as desired Proof of Proposition 7: Without loss of generality, we may suppose that N = k We prove the result by induction over k Suppose that Ω has k+ atoms Apply Lemma 8 to cover Ω by k subsets, and to give x, x,, x S, y, y,, y S, z, z,, z S as described in the lemma Then, by the triangle inequality S S S S x s, x s, + y s, + z s, By the induction hypothesis, x s, c k 7

8 THE RADEMACHER COTYPE OF OPERATORS FROM l N By Lemmas and we have that S y s, c and Hence z s, c c 4 k x s, as required, taking c = + c c 4 To prove the main result is now easy ) + log k c c 4 c k + ), Proof of Theorem : By Theorem 5, it is sufficient to show that for any probability measure µ on {,,, N}, the cotype constant of the canonical embedding l N L, µ) is bounded by some universal constant times log log N But this is precisely what Proposition 7 says Final Remarks There is a similar result for Gaussian cotype see [Mo]) Theorem There is a constant c such that, for any operator T : l N Y, where Y is a Banach space, the Gaussian cotype constant, β ) T ), is bounded according to the relation: β ) T ) c log log N π, T ) This result is the best possible, as is implicitly shown in [T] Theorem 3 There is a constant c such that for any integer N, there is an operator T : l N Y, where Y is a Banach space, such that β ) T ) c log log N π, T ) Since the Rademacher cotype constant is greater than a constant times the Gaussian cotype constant, we have the following corollary 8

9 MONTGOMERY-SMITH TALAGRAND Corollary There is a constant c such that for any integer N, there is an operator T : l N Y, where Y is a Banach space, such that K ) T ) c log log N π, T ) We also have the following, the result originally stated in [T] Corollary There is an operator T : CK) Y, where Y is a Banach space, that is, )-summing, but does not have Rademacher cotype If we write R N for the supremum of K ) T )/π, T ) over all T : l N Y, then we have shown that c log log N R N c log log N Clearly, we are left with the following problem Open Question What is the asymptotic behavior of R N? Acknowledgements The main result of this paper originally appears in the PhD thesis of the first named author [Mo], and he would like to express his thanks to his advisor, DJH Garling, and to the Science and Engineering Council who financed his studies He would also like to express his gratitude to GJO Jameson who first suggested the problem to him, and gave him much encouragement 9

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