A CLASS OF SCHUR MULTIPLIERS ON SOME QUASI-BANACH SPACES OF INFINITE MATRICES

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1 A CLASS OF SCHUR MULTIPLIERS ON SOME QUASI-BANACH SPACES OF INFINITE MATRICES NICOLAE POPA Abstract In this paper we characterize the Schur multipliers of scalar type (see definition below) acting on scattered classes of infinite matrices In [9], J Schur introduced a new product between two matrices A = (a jk ) and B = (b jk ) of the same size, finite or infinite This product, known in the literature as the Schur product or Hadamard product, is defined to be the matrix of elementwise products A B = (a jk b jk ) This concept was used in different areas of Analysis as complex function theory, Banach spaces, operator theory and multivariate analysis G Bennett studied in [4] the behaviour, under Schur multiplication, of the norm p,q, 1 p, q, 1/q A pq = sup x p 1 j q a jk x k In particular, he was interested in characterizing the (p,q)-multipliers: the matrices M for which M A maps l p into l q whenever A does In his paper it is proved a theorem about Schur multipliers which are Toeplitz matrices, that is about the matrices of the form: a 0 a 1 a 2 a 3 a 1 a 0 a 1 a 2 A = a 2 a 1 a 0 a 1 a 3 a 2 a 1 a 0 where (a j ) j= is a sequence of complex numbers Theorem 81 -[4] reads: Teorem BA Toeplitz matrix A is a Schur multiplier if and only if µ = j= a je ijt is a bounded Borel measure on [0, 2π) This fact leads naturally to the idea of identifying the Schur multipliers with the noncommutative bounded Borel measures, see eg [3] We denote by M(l 2 ) the space of all (2,2) Schur multipliers from B(l 2 ) into B(l 2 ), where B(l 2 ) is, as usual, the Banach space of linear and bounded operators on l 2 with the usual operator norm 2000 Mathematics Subject Classification 42A16,42A45, 47B35 Key words and phrases Schur multipliers, matrices of scalar type 1 k

2 2 NICOLAE POPA The space M(l 2 ) endowed with norm A M(l2) = sup A B B B(l2 ) 1 B(l 2) becomes a Banach space Since we work with different quasi-banach spaces of matrices X, Y we use the notation (X, Y ) for the space of all Schur multipliers from X into Y equipped with the quasi-norm A (X,Y ) = sup B Y 1 A B X In this way (X, Y ) becomes a quasi Banach In [4] G Bennett raised the problem of characterizing the Hankel matrices which are Schur multipliers We recall that a matrix A is called a Hankel matrix if it is defined by a sequence (a j ) j=1 of complex numbers in the following way: a 0 a 1 a 2 a 3 a 1 a 2 a 3 a 4 A = a 2 a 3 a 4 a 5 a 3 a 4 a 5 a 6 G Pisier in [8], solved the above problem He proved the following theorem Theorem P A Hankel matrix is a Schur multiplier if and only if the Fourier multiplier n=0 x ne int n=0 a nx n e int maps boundedly H 1 (S 1 ) into itself Here H 1 (S 1 ) is the Hardy space of the Schatten class S 1 valued analytic functions, endowed with the norm f H 1 (S 1) = 1 2π 2π 0 n=0 A ne int S1 dt < For the definition of the Schatten classes S p see eg [1] In [1], A B Alexandrov and V V Peller characterized the Toeplitz matrices which are Schur multipliers for S p, 0 < p < 1 They proved the following theorem Theorem AP Let 0 < p < 1 A Toeplitz matrix T given by the complex sequence (t j ) j= belongs to (S p, S p ) if and only if there exists a measure µ M p with the Fourier coefficients µ(j) = t j Moreover, in this case T (Sp,S p) = µ Mp, where M p = {µ : T C µ = j α jδ tj, t j T, t j distinct points }, µ Mp = ( j α j p ) 1/p < and δt is the Dirac measure concentrated at the point t T The above mentioned papers [1,8] show that a complete description of general Schur multipliers, at least, either for B(l 2 ) or S p, 0 < p 1, is a difficult target In this way it is natural to consider and study other classes of Schur multipliers than those which are Toeplitz matrices In [2], the following notation, more apropriate for our aims, for the entries of a matrix B was introduced Namely, we put { b l bl,l+k, k 0, l = 1, 2, k = b l k,l, k < 0, l = 1, 2, and write B = (b l k ) l 1, k Z Let B (l) = (b m k ) k Z, m 1, where l = 1, 2, 3,, be the matrix given by { b m b l k = k, if m = l 0 if m l

3 A CLASS OF SCHUR MULTIPLIERS 3 We call the matrix B (l), the lth corner matrix associated to B Now, we associate to each matrix B l a periodical distribution on T, denoted by f l, such that b l k = f l (k) and we identify the matrix B = ( B (l)) with the l N sequence of associated distributions (f l ) l N Then for the sequence α = (α 1, α 2, ) and the matrix B = (f l ) l N, we denote by α B the matrix given by ( α l f l )l N In particular, if B is a Toeplitz matrix (B T, ) and if α is the constant sequence then α B coincides with the matrix αb Hence, if [α], is the matrix α 1 α 1 α 1 α 1 α 2 α 2 [α] = α 1 α 2 α 3 it is clear that α B = [α] B We define ms to be the space of all sequences α such that α B B(l 2 ) for all B B(l 2 ), or equivalently [α] M(l 2 ) On ms we consider the norm α ms = [α] M(l2) Then ms is a unital commutative Banach algebra with respect to the usual multiplication of sequences As it was observed in [2], the multiplication of a function with a scalar corresponds to the multiplication of a sequence and an infinite matrix We call the matrices [α], scalar matrices In this context, in [2] a theorem of Haar s type for infinite matrices was proved The product appeared also in [6] in other contexts An important role in applications is played by the upper triangular projection applied to the matrix [α] For an infinite matrix A = (a ij ) i 1, j 1, the upper triangular projection is { ai,j, if i j P T (A) = 0, otherwise A sequence b = (b n ) n 1 belongs to pms if and only if B = {b} = P T ([b]) M(l 2 ) The space pms endowed with the norm b = {b} M(l2) becomes a Banach algebra with respect to the usual product of sequences In [2] there were given sufficient and necessary conditions in order that matrices of the form [α] or {α}, ie α 1 α 1 α 1 0 α 2 α 2 {α} = 0 0 α 3 to be Schur multipliers The following result was proved Theorem BLP1 Let b = (b n ) n 1 be a complex sequence

4 4 NICOLAE POPA 1) If (i n ) n 1 is a strictly increasing sequence of natural numbers with i 1 = 0, and z in = max i n<k i n+1 b k, then there is a constant R > 0 such that {b} M(l2) R inf { (z in ) n (z in log(i n+1 i n )) n } (i n) n 1 2) If b pms then (log n) 2 n+p sup b k 2 < n 1; p 1 n 3) If ( b k ) k 1 is a decreasing sequence, then b pms if and only if b k = O(1/ log k) k=p As an immediate consequence we have: Corollary 1 1) l 2 ms l 2) {(b n ) n 1 b n = O(1/ log n)} ms A set of sufficient conditions in order that a matrix of the type [α] to be a Schur multiplier is given in [2], namely the following theorem was proved Theorem BLP2 Let b = (b n ) n 1 a complex sequence Then: 1) If sup n 1 n j=1 b j b n 2 <, then b ms 2) If b BV N = b 1 + n=1 b n+1 b n < then b ms It is well-known that M(l 2 ) coincides with (S 1, S 1 ), the space of all Schur multipliers from S 1 into S 1, see e g [8] Using this fact we give a simpler proof of the first statement of Corollary 1 Theorem 2 Let c = (c n ) n 1 l 2 Then c pms Proof By using the Schmidt decomposition of a matrix A, it is enough to show that A {c} S 1 for a matrix A of rank 1 Let A = α β with α = (α n ) n 1 l 2 and β = (β n ) n 1 l 2 We have c 1 c 1 c 1 c 1 α 1 β 1 α 1 β 2 α 1 β 3 α 2 β 1 α 2 β 2 α 2 β 3 A {c} = α 3 β 1 α 3 β 2 α 3 β 3 0 c 2 c 2 c c 3 c 3 c 1 α 1 β 1 c 1 α 1 β 2 c 1 α 1 β 3 0 c 2 α 2 β 2 c 2 α 2 β 3 = 0 0 c 3 α 3 β 3 By the definition of S 1 and Cauchy-Schwartz inequality we get ( ) A {c} S1 (α j c j β j+k ) k=0 l2 = α j c j 2 β j+k 2 j=1 j=1 k=0 α j c j β j l2 α l2 β l2 c l2 = c l2 A S1, j=1

5 A CLASS OF SCHUR MULTIPLIERS 5 that is A M(l2) c l2 and the proof is complete We characterize now the upper triangular scalar matrices which are Schur multipliers, from the Hardy space H 2, respectively from the Schatten class S 2 into B(l 2 ) Theorem 3 1) Let H 2 be the Hardy space of Toeplitz matrices generated by the classical Hardy space of functions Then an upper triangular matrix A = {α} belongs to (H 2, B(l 2 )) if and only if α l 2 Moreover, we have equality of the norms 2) Let T 2 be the space of all upper triangular Hilbert-Schmidt matrices Then {α} (T 2, B(l 2 )) if and only if α l Proof 1) We use the following identity proved in [2] 1 2 (1) B B(l2) = sup b kj e 2πijt h( t)dt, h 2 1; h H0 2[0,1] 0 k=1 where B is an upper triangular matrix B = (b kj ) Then, if f(t) = k=0 c ke 2πikt H 2, t [0, 1], F is the Toeplitz matrix associated to f (that is F is given by (c k ) k 0 ), and α l 2, we have: α 0 c 0 α 0 c 1 α 0 c 2 α 0 c 3 0 α 1 c 0 α 1 c 1 α 1 c 2 F {α} = 0 0 α 2 c 0 α 2 c, F {α} B(l2) = sup α k 1 c j e 2πijt h( t)dt = h 2 1 k=1 0 j=0 ( ) 1 sup α k 1 2 c j e 2πijt h( t)dt h = α l 2 f H 2 k=1 j=0 Hence {α} (H2,B(l 2)) = α l2 and this completes the proof c 11 c 12 c 13 0 c 22 c 23 2) Let α l and C = 0 0 c 33 T 2Using the formula (1) and Cauchy-Schwartz inequality we get 1 {α} C B(l2) α k 1 2 sup 2 c kj e 2πijt h( t) h j k k=1 j=k α k 1 2 c kj 2 j k k=1

6 6 NICOLAE POPA ( sup k 0 α k ) c kj 2 k=1 j k α l C T2 Hence {α} (T 2, B(l 2 )) and this proves the first part of the theorem Conversely, let {α} (T 2, B(l 2 )), that is C {α} B(l 2 ) for all C T 2 and take C = C 0, that is the matrix C is reduced to main diagonal It is clear that the sequence of entries of this diagonal belongs to l 2 Consequently the sequence (α k 1 c kk ) k 0 belongs to l for every sequence (c kk ) k 1 l 2 Hence (α k ) k=0 l and the proof is complete Next we use the important results of G Bennett proved in [5], in order to characterize the Schur multipliers of scalar type for some spaces of lower triangular infinite matrices contained in the Schatten classes S p, 0 < p < We denote these spaces by LT S p Next we get a general description of upper triangular Schur multipliers of scalar type for different quasi-banach spaces In order to state the following result we need to recall some definitions (see [10]) Let f be the space of all sequences with a finite number of non-zero elements A norm Φ on f is called symmetric if Φ(a) = Φ(a ), for all a f, that is if Φ is invariant to permutations and to applications a n e iθn a n, where θ n is a sequence of real numbers Here a = (a n) n=1 is the decreasing rearrangement of the sequence (a n ) which converges to 0 We say that the sequence (a n ) n belongs to the space s Φ, if and only if lim n Φ(a 1,, a n, 0, 0, ) = Φ(a) exists We denote by S Φ the space of all compact operators A on l 2 with the sequence of their singular numbers (µ n (A)) belonging to s Φ For A S Φ we put Φ(A) = Φ((µ n (A)) n ) Then the following non-commutative Hölder type inequality proved in [10] holds Theorem AH Let Φ 1, Φ 2, Φ 3 be symmetric norms such that if a s Φ2, b s Φ3 then ab s Φ1 and Φ 1 (ab) Φ 2 (a)φ 3 (b) If A S Φ2, B S Φ3, then AB S Φ1 = and Φ 1 (AB) Φ 2 (A)Φ 3 (B) Using this inequality we can state the following interesting result Theorem 4 Let s Φ1 = s Φ2 s Φ3 (ie for each α s Φ1, there exist β s Φ2, γ s Φ3 such that α = βγ, and Φ 1 (α) inf α=βγ Φ 2 (β)φ 3 (γ)) Then a scalar matrix [α] (LT S Φ2, LT S Φ1 ) if and only if α s Φ3 Proof Let first A LT S Φ1 and α s Φ3 Then it is clear that where A [α] = A D α, α α 2 0 D α = 0 0 α 3

7 A CLASS OF SCHUR MULTIPLIERS 7 By Theorem AH it follows that A [α] SΦ1 A D α SΦ1 A SΦ2 D α SΦ3 A SΦ2 α sφ3 Hence [α] (LT S Φ2, LT S Φ1 ) and this completes the first part of the proof For the reverse implication, take A to be the main diagonal with the entries (a jj ) j=1 s Φ 2 and [α] (LT S Φ2, LT S Φ1 ) a 11 α a 22 α 2 0 Then A [α] = A D α = 0 0 a 33 α S Φ1 and we get that 3 (a ii α i ) i=1 s Φ 1 for all sequences (a jj ) j=1 s Φ 2 Since s Φ1 = s Φ2 s Φ3, it follows that α s Φ3 and this completes the proof of the theorem Let w = (w n ) be a positive decreasing sequence of numbers Of course the Lorentz space of sequences l p,w, 0 < p, is a space of the previous type s Φ, see eg [10] By the well-known fact that l p,w l q,w = l r,w, for 1 p + 1 q = 1 r ; 0 < p, q, r < we get the following result Corollary 5 1) Let 1 p + 1 q = 1 r ; 0 < p, q, r < Then [α] (S p, S r ) if and only if α l q 2) Let w n be a decreasing positive sequence and 0 < p, q <, be such that 1 p + 1 q = 1 Then [α] (S p,w, S 1,w ) if and only if α l q,w, where l p,w is the weighted Lorentz space of sequences We call the Bergman-Schatten space of order p, 0 < p < and we denote by L p a(l 2 ) the space of all upper triangular matrices A such that A L p a (l 2) = ( 1 0 k=0 A kr k p 1/p S p 2rdr) < See eg [7] for further notations and details By Hölder s inequality we get the following result Theorem 6 Let 1 p < Then [α] ( L p a(l 2 ), L 1 a(l 2 ) ) if and only if α l q, where 1 p + 1 q = 1 Proof Let A L p a(l 2 ) and α l q We clearly have that A [α] = D α A By Theorem AH we get ( 1 A [α] L 1 a (l 2) = D α A(r) S1 2rdr A(r) p S p 2rdr) 1/p α lq = A L p a(l 2) α lq, ie [α] ( L p a(l 2 ), L 1 a(l 2 ) ) and this completes the first part of the proof Conversely, let [α] ( L p a(l 2 ), L 1 a(l 2 ) ) By taking A = A 0 = (a jj ) j=1 Lp a(l 2 ), ie for (a jj ) j l p, we get A [α] = α 1 a α 2 a 22 L1 a(l 2 ),

8 8 NICOLAE POPA or, equivalently, (α j a jj ) j l 1 Hence by Hölder s inequality it follows that (α j ) j l q and the proof is complete Using the results of G Bennett, proved in [5] we can also describe the Schur multipliers of scalar type also for others quasi-banach spaces of matricesthe spaces of sequences d(a, p), g(a, p) and ces(p) were defined in [5] We denote now by d q M (a, p), gq M (a, p), cesq M (p) and lq M (p) the spaces of upper triangular infinite matrices A = k=0 A k, with all the sequences on the diagonals belonging to d(a, p) (respectively g(a, p), ces(p), l p, ) and such that A = ( k A k q d(a,p) ) 1/q < (respectively A = ( k A k q g(a,p)) 1/q < and so on) with the usual modification for q = Using Theorems 45 and 38 - [5], we have: Theorem 7 1) Let 1 < p < Then α g(p ), where 1 p + 1 p = 1, if and only if [α] (l M (p), ces M (p)), where g(p ) = g(a, p ), with a = (1, 1, ) 2) Let 0 < p < Then [α] (d M (a, p), l M (p)) if and only if α g(a, p) Theorem 8 1) For 1 < p <, [α] (l q M (p), cesq M (p)) if and only if α g(p ) 2) For 0 < p <, [α] (d q M (a, p), lq M (p)) if and only if α g(a, p) Acknowledgement 1) This paper was partially supported by the CNCSIS grant ID-PCE 1905/2008 2) The author thanks to the referee for his/her valuable remarks which improve considerable the presentation of this paper References [1] A B Alexandrov, V V Peller, Hankel and Toeplitz-Schur multipliers, Math Ann, 324(2002), [2] S Barza, V Lie, N Popa, Approximation of infinite matrices by matricial Haar polynomials, Ark Mat, 43(2005), [3] S Barza, Lars-Erik Persson, N Popa, A matriceal analogue of Fejer s theory, Math Nachr, 260(2003), [4] G Bennett, Schur multipliers, Duke Math J, 44(1977), [5] G Bennett, Factorizing the classical inequalities, Memoirs Amer Math Soc,120(1996), [6] A Marcoci, L E Persson, N Popa, Schur multiplier characterization of a class of infinite matrices, Czech Math J, 60(2010), [7] L G Marcoci, L E Persson, I Popa, N Popa, A new characterization of Bergman-Schatten spaces and a duality result, J Math Anal Appl, 360(2009), [8] G Pisier, Similarity problems and completely bounded maps, LNM 1618, Springer-Verlag, Berlin, 1995 [9] J Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J Reine Angew Math, 140(1911), 1-28 [10] B Simon, Trace ideals and their applications, LNM 35, Cambridge Univ Press, Cambridge, 1979 Institute of Mathematics of Romanian Academy, Unit research 1, PO BOX RO Bucharest, ROMANIA npopa@imarro