ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S XX) ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS WILLIAM D. BANKS AND ASMA HARCHARRAS Abstract. We show that if the norm of an idemotent Schur multilier on the Schatten class S lies sufficiently close to 1, then it is necessarily equal to 1. We also give a simle characterization of those idemotent Schur multiliers on S whose norm is Introduction We study norms of idemotent Schur multiliers defined on the Schatten -class with 1 < <,. For any idemotent Schur multilier φ, we show that if the norm of φ lies sufficiently close to 1, then it is necessarily equal to 1. More recisely, if φ is an idemotent Schur multilier on the Schatten -class, then φ = 0, φ = 1, or φ 1 + η, where η is a ositive constant that deends only on. We also obtain a simle characterization of those idemotent Schur multiliers whose norm is equal to 1. When = 1 or, these results have been obtained by Livshits [], while for =, every nonzero idemotent Schur multilier has norm 1. To state our results more exlicitly, we need to fix some standard terminology. For every real number in the range 1 <, denote by S the Schatten -class over the Hilbert sace l ; it is the Banach sace of all comact oerators x : l l with finite norm x S = Tr x x) /) 1/, where Tr ) denotes the usual trace. For =, the sace S is the Banach sace of all comact oerators x : l l, equied with the usual oerator norm. The saces S, 1, were considered in [4] as noncommutative analogues for the saces l, 1 for a more modern reference, see [3] for examle). For 1 and a ositive integer n, let Sn denote the Schatten -class over the Hilbert sace l n of dimension n. In what follows, we make no distinction between an oerator x on l and the corresonding matrix x ij ) i,j N relative to the canonical basis {e ij } i,j N of S. A set-theoretic ma φ : N N C is said to be a Schur multilier on S if the associated oerator T φ : S S, defined by T φ x) = φ ij x ij ) i,j N, x = x ij ) i,j N S, is well defined and bounded on S. In articular, this imlies that φ itself is a bounded ma. Let MS ) denote the sace of all Schur multiliers on S. Then 1991 Mathematics Subject Classification. Primary 47A30; Secondary 47B49, 47B10. Key words and hrases. Schur multilier, Schatten class. The first author was suorted in art by NSF grant DMS c 1997 American Mathematical Society

2 WILLIAM D. BANKS AND ASMA HARCHARRAS MS ) is a Banach algebra when it is equied with the ointwise roduct and the norm φ MS ) = T φ : S S, φ MS ). It is well known that for airs 1, q with 1 +q 1 = 1, the algebras MS ) and MS q ) can be identified isometrically. These identifications can be done via the identity ma by defining the duality between S and S q with x, y = Tr t xy) for all x S and y S q. In addition, the sace MS ) can be identified isometrically with the Hilbert sace l N N). Consequently, when studying MS ) it suffices to reduce to the case where <. Finally, a Schur multilier φ MS ) is said to be idemotent rovided that T φ T φ = T φ ; clearly, this is equivalent to the condition that φ mas N N into the set {0, 1}. For such multiliers, one has φ MS ) = φ φ MS ) φ MS ). Hence, φ MS ) 1 whenever φ 0. Our main result is the following: Theorem 1.1. For every real number with 1 < < and, there exists a constant η > 0 deending only on ) such that for every nonzero idemotent Schur multilier φ MS ) with φ MS ) 1, the following inequality holds: φ MS ) 1 + η. By the remarks above, it suffices to consider the case where < <, which we assume throughout the sequel. Acknowledgements. We thank G. Pisier for bringing this question to our attention; the question was insired by a talk given by V. Paulsen at the AMS Conference held in Irvine during the Fall of Proof of the Main Result The roof of Theorem 1.1 can be slit into three ieces, as follows. Lemma.1. Let = ij ) 1 i,j with 11 = 1 = = 1 and 1 = 0. Then MS ) > MS ) > 1 for < <. Proof. For every c C, let x c) = x c) ij ) 1 i,j, where x c) 11 = xc) 1 = xc) = c. One has x c) 1 x c) S = Tr x c) x c)) / ) 1/ = λ / +,c + λ /,c) 1/, where λ ±,c = c ± ) Rc) + c + c 4. In articular, if we choose c = )/, then where f) is the function MS ) xc) ) S x c) S = x0) S x c) S = f), = 1 and 3 + 5) / + 3 5) /) ) + 16 ) / ) + 16 ) /

3 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER 3 Since f) = 1, f ) = 0, and f ) = log3 + 5) 3 5 log3 5) > 0, the Taylor exansion for f) near = shows that f) > 1 if < < + ε, for some ε > 0. Thus, MS ) > 1, < < + ε. Now let > > be arbitrary real numbers and let 0 < θ < 1 be chosen such that 1/ = 1 θ)/ + θ/. By the classical results of comlex interolation, we have S = S, ) S isometrically for the definition and fundamental results on θ comlex interolation, the reader is referred to [1]), hence it follows that MS ) 1 θ MS ) θ MS ). Taking < < + ε with ε sufficiently small, and using the obvious fact that MS ) = 1, the receding relation and our results above imly that MS ) > 1 for all <. Since 0 < θ < 1, the above relation further imlies that MS ) > MS ) for < <. This comletes the roof. It has been shown in [] that MS ) = / 3, which rovides an uer bound for MS ) for any >. On the other hand, in the notation of Lemma.1 and taking c = 1, we have for > : MS ) x0) S 3 + 5) / + 3 ) 1/ 5) / = x 1) S +1 >. 1+1/ It remains an interesting question to determine the recise value of MS ) for any in the range < < ; this will not be needed, however, in what follows. We now define, for each in the range < <, η = 1 + MS ). In view of Lemma.1, η is strictly ositive. Definition.. A ma φ defined on N N or any of its subsets) is said to be triangle-free if there are no integers i, j, k, l such that φ ij = φ il = φ kj = 1 and φ kl = 0. The following lemma is an easy consequence of Lemma.1; the roof is omitted. Lemma.3. Fix >, and suose that φ MS ) is idemotent. If then φ is triangle-free. Finally, we have: φ MS ) < 1 + η, Lemma.4. If a ma φ : N N {0, 1} is nonzero and triangle-free, then φ MS ) = 1 for every real number >. Proof. For any ositive integer n, denote by φ n) the restriction of φ to the subset {1,,..., n} {1,,..., n} of N N. Recalling the well known fact φ MS ) = su φ n) MS n ), n 1

4 4 WILLIAM D. BANKS AND ASMA HARCHARRAS we see that it suffices to show that φ n) MS n ) = 1 whenever φ n) 0. To this end, let n 1 be fixed with φ n) 0. For every integer 1 i n, define the row sum c i = #{1 j n φ n) ij = 1}. To show φ n) MS n ) = 1, we may freely ermute the rows and/or the columns of φ n) in any way that we want; in articular, without loss of generality, we may assume that c 1 c c 3... c n, and that φ n) 11 = φn) 1 = φn) 13 =... = φn) 1c 1 = 1. Since φ is triangle-free, for every 1 i n there are only two ossibilities: α) φ ij = 1 for all 1 j c 1, and φ ij = 0 for all j > c 1, β) φ ij = 0 for all 1 j c 1. After ermuting the rows if necessary, we may assume that α) occurs for 1 i r 1, and that β) occurs for i > r 1. Then φ n) = φ 1 φ 1, where φ 1 is an r 1 c 1 rectangular matrix with every entry equal to 1, and φ 1 is an n r 1 ) n c 1 ) rectangular matrix whose entries are equal to 0 or 1 and which is triangle-free. If φ 1 = 0, we sto; otherwise, we reeat the same argument with φ n) relaced by φ 1, obtaining φ n) = φ1 φ φ. We continue in this way until the rocess stos, at which oint we have φ n) = φ 1 φ... φ s, where every φ k, 1 k s, is an r k c k rectangular matrix, all of the entries of φ 1,..., φ s 1 are equal to 1, and the entries of φ s are all equal to 1 or all equal to 0. By adding some additional zero rows and/or zero columns to φ n) if necessary, we may also assume that r k = c k for 1 k s. Then and the result follows. φ n) MS n ) = su φ k MS rk ) = 1, 1 k s Theorem 1.1 is an immediate consequence of Lemmas.1.4, as the reader can easily verify. Examining the roof of Theorem 1.1, we see that for a nonzero idemotent Schur multilier φ, the following assertions are equivalent: a) For some >, φ : S S has norm 1, b) φ is triangle-free, c) φ is equivalent to a multilier of the form φ 1 φ φ 3..., where each φ j has all of its entries equal to 1 or all of its entries equal to 0, d) φ : S S has norm 1, e) For every, φ : S S has norm 1.

5 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER 5 References [1] J. Bergh, J. Löfström, Interolation saces: an introduction, Sringer-Verlag, Berlin-New York, [] L. Livshits, A note on 0-1 Schur multiliers, Linear Algebra Al. 1995), 15. [3] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint oerators, Translations of Mathematical Monograhs, Vol. 18, American Mathematical Society, Providence, R.I., [4] R. Schatten, Norm ideals of comletely continuous oerators, Sringer-Verlag, Berlin- Göttingen-Heidelberg 1960). Deartment of Mathematics, University of Missouri, Columbia, MO 6511 USA address: bbanks@math.missouri.edu Deartment of Mathematics, University of Missouri, Columbia, MO 6511 USA address: harchars@math.missouri.edu