Table Of Content

SCHUR MULTIPLIERS AND OPERATOR-VALUED FOGUEL-HANKEL OPERATORS 5 0 0 C. BADEA AND V.I. PAULSEN 2 n a Abstract. WeshowthatsomematricesareSchurmultipliersand J this is applied to obtain classes of operator-valued Foguel-Hankel 1 operators similar to contractions. This provides partial answers 1 to a problem of K. Davidson and the second author concerning ] CAR-valued Foguel-Hankel operators. A F . h t a m 1. Introduction [ AnexampleofapolynomiallyboundedoperatoronHilbertspacenot 1 v similar to a contraction was found recently by Pisier [Pi]. An operator- 0 theoretic proof that certain CAR-valued Foguel-Hankel operators are 5 1 polynomially bounded operators but not similar to contractions was 1 0 givenbyDavidsonandPaulsen[DP]. Itisstillanopenquestion[DP]to 5 characterize operators in this family which are similar to contractions. 0 / The aim of this note is to prove some partial results concerning h t a this open problem. The present note is a sequel of [DP] where this m problemisstudied. Acertainfamiliaritywith[DP]issupposed. Forthe : v convenience of the reader some notation and known facts are recalled i X below. r a 1.1. Background. We denote by H a separable Hilbert space and by B(H) the C*-algebra of all bounded and linear operators on H. An operator T ∈ B(H) is said to be power bounded if (1.1) sup kTnk < +∞ n∈Z+ 1991 Mathematics Subject Classification. 47A20, 47A56, 47B35. Key words and phrases. Similarity to contraction, Foguel-Hankel operators, Schur multipliers. 1 2 C. BADEA ANDV.I. PAULSEN and polynomially bounded if there exists a constant K such that (1.2) kp(T)k ≤ Kkpk ∞ for each analytic polynomial p. Here kpk := sup{|p(z)| : |z| ≤ 1}. ∞ We say that T is similar to a contraction if there is an invertible operator L ∈ B(H) such that kL−1TLk ≤ 1. The following implications hold : T similar to a contraction ⇒ T polynomially bounded (1.3) ⇒ T power bounded . The first implication follows from von Neumann’s [vN] inequality kp(C)k ≤ kpk , ∞ valid for each contraction C ∈ B(H). The second implication is clear from (1.1) and (1.2). It was proved by Paulsen [Pa1] that T ∈ B(H) is similar to a contraction if and only if T is completely polynomially bounded, that is, there exists a constant K such that k[p (T)] k ≤ Ksup{k[p (z)] k : |z| ≤ 1}, ij 1≤i,j≤n ij 1≤i,j≤n for all positive integers n and all n×n matrices [p ] with poly- ij 1≤i,j≤n nomial entries. Recall that [p (T)] is identified with an operator ij 1≤i,j≤n acting on the direct sum of n copies of the corresponding Hilbert space in a natural way. No implication in (1.3) can be reversed. The first power bounded operator not similar to a contraction was constructed by Foguel [F]. His counterexample has the form S∗ X (1.4) R(X) = R(S∗,S;X) = , 0 S " # FOGUEL-HANKEL OPERATORS 3 whereS denotestheunilateralshiftonℓ andX wasasuitablediagonal 2 projection onto a subspace of ℓ . We will call Foguel operators the 2 operators of type (1.4). Lebow [L] proved that the example constructed by Foguel is not polynomially bounded. Other examples of power bounded, not polynomially bounded operators are in [Da, Pe2, Bo]. A Foguel-Hankel operator is a Foguel operator (1.4) with X = [a ] = Γ , i+j i,j≥0 f a Hankel operator with symbol f (cf. [DP]). The study of Foguel- Hankel operators was initiated by Foias and Williams [FW] and Peller [Pe1]. It follows from the work of Peller [Pe1], Bourgain [B] and Alek- sandrov and Peller [AP] that these Foguel-Hankel operators are similar to contractions whenever they are polynomially bounded. Both conditions are equivalent to f′ ∈ BMOA , that is with the boundedness of Γf′ = [(i+j +1)ai+j]i,j≥0. However, it is still unknown if a general Foguel operator is similar to a contraction whenever it is polynomially bounded. See [Fe], where this is related to computing a certain Ext group. The first example of a polynomially bounded operator not similar to a contraction was found by Pisier [Pi]. His example is a CAR-valued Foguel-Hankel operator which we introduce below. 1.2. CAR-valued Foguel-Hankel operators. Let Λ be a function from H into B(H) satisfying the CAR - canonical anticommutation relations : for all u,v ∈ H, Λ(u)Λ(v)+Λ(v)Λ(u) = 0 and Λ(u)Λ(v)∗+Λ(v)∗Λ(u) = (u,v)I. The range of Λ is isometric to Hilbert space. Let {e } be an or- n n≥0 thonormal basis for H, and let C = Λ(e ) for n ≥ 0. For an arbitrary n n 4 C. BADEA ANDV.I. PAULSEN sequence α = (α ,α ,...) in ℓ2, let 0 1 Y = α C α i+j i+j h i be a CAR-valued Hankel operator. Let S∗(∞) Y R(Y ) = R(S∗(∞),S(∞);Y ) = α α α 0 S(∞) " # be the corresponding CAR-valued Foguel-Hankel operator [Pi], [DP]. The initial choice of α made by Pisier was α = 1 for k ≥ 0 and 2k−1 α = 0 otherwise. In this case R(Y ) is polynomially bounded but not i α completely polynomially bounded. The following more general result holds (cf. [Pi], [DP]). For a fixed sequence α = (α ,α ,...) ∈ ℓ , let 0 1 2 A = A(α) := sup(k +1)2 |α |2 i k≥0 i≥k X and B = B (α) := (k +1)2|α |2. 2 2 k k≥0 X The operator R(Y ) is polynomially bounded if and only if A is finite. α If R(Y ) is similar to a contraction, then B is finite. α 2 It is an open problem [DP] to characterize in terms of the sequence α when R(Y ) is similar to a contraction. In particular, it is not known α if B (α) finite implies the similarity of R(Y ) to a contraction. Note 2 α [DP, p. 163] that B2(α) = kΓF′k2 = k(i+j +1)αi+jCi+jk2, where the operator-valued symbol F given by F(z) = α C z−n−1 n n n≥0 X is such that Γ = Y . F α We refer to [Pi], [DP], [D]for moreinformationandfor the undefined terms. FOGUEL-HANKEL OPERATORS 5 1.3. Organization of the paper. The main results are stated in the next section. Section three contains some useful results about Schur multipliers. In the fourth section, a sufficient condition for similarity is given. These results are used to prove the main results in the last section. 2. Main results We use notations as above. The first two results give sufficient conditions for similarity to a contraction of an operator-valued Foguel operator. Although these results are implicit in the work of [FW] and [DP], they do not seem to have been stated elsewhere. Theorem 2.1. Let X ∈ B(ℓ (H)) with matrix X = [X ] , X ∈ 2 ij i,j≥0 ij B(H), with respect to a fixed orthonormal basis. For each n ≥ 1 set A(n)(X) = X +X +...X ij ij i−1,j+1 i−min(i,n−1),j+min(i,n−1) and let A(n)(X) be the matrix [A(n)(X) ] . If ij ij i,j≥0 (2.1) supkA(n)(X)k < +∞, n≥1 then the operator-valued Foguel operator S∗(∞) X R(S∗(∞),S(∞);X) = ∈ B(ℓ (H)⊕ℓ (H)) 0 S(∞) 2 2 " # is similar to a contraction. In the case of operator-valued Foguel-Hankel operators the following holds. Theorem 2.2. Let Γ = [Γ ] be an operator-valued Hankel oper- i+j i,j≥0 ator, Γ ∈ B(H). The operator-valued Foguel-Hankel operator k S∗(∞) Γ R(S∗(∞),S(∞);Γ) = ∈ B(ℓ (H)⊕ℓ (H)) 0 S(∞) 2 2 " # is similar to a contraction if any of the following operators ΓD−D∗Γ = [(j −i)Γ ] , i+j−1 i,j≥0 ΓD = [jΓ ] , i+j−1 i,j≥0 6 C. BADEA ANDV.I. PAULSEN or D∗Γ = [iΓ ] i+j−1 i,j≥0 is a bounded operator. Here Γ = 0 and D = [(i+1)δj ] is the −1 i+1 i,j≥0 differentiation operator. If Γ = Γ = (a ) is a scalar Hankel operator with symbol f, f i+j then boundedness of any one of the three operators above is equivalent to boundedness of the other two and this occurs if and only if f′ ∈ BMOA. This is a consequence of the fact that both conditions are equivalent to similarity to a contraction of the Foguel-Hankel operator. This, as was remarked in [AP], also follows from [JP]. Foroperator-valuedHankels, thesituationismuchmorecomplicated asisshownin[DP]. Asufficient conditionforsimilaritytoacontraction foranoperator-valuedFoguel-HankeloperatorwasgivenbyBlower [Bl] in terms of Carleson measures. In the case of CAR-valued Foguel-Hankel operators, we still do not know if B (α) < +∞ implies the similarity to a contraction of R(Y ). 2 α Theorem 2.2 and the Schur multipliers results of the next section will imply the following results. Theorem 2.3 (loglog condition). Let ε > 0. Suppose (k +1)2[log(k +1)]2[log(log(k +1))]2+ε|α |2 < +∞. k k≥1 X Thenthe CAR-valuedFoguel-HankeloperatorR(Y ) = R(S∗(∞),S(∞);Y ) α α is similar to a contraction. Since the logarithm goes to infinity less quickly than any power, we obtain the following consequences. Corollary 2.4 (log condition). Let ε > 0. Suppose (k +1)2[log(k +1)]2+ε|α |2 < +∞. k k≥0 X Then the CAR-valued Foguel-Hankel operator R(Y ) is similar to a α contraction. and FOGUEL-HANKEL OPERATORS 7 Corollary 2.5 (power condition). Let ε > 0. Suppose B (α) := (k +1)2+ε|α |2 < +∞. 2+ε k k≥0 X Then the CAR-valued Foguel-Hankel operator R(Y ) is similar to a α contraction. A proof of Corollary 2.5 can be given by combining results from [DP] and [P]. A different proof in the case ε = 1 can be found in [Ba]. 3. Schur multipliers Let A = [a ] and B = [b ] be two matrices of the same size ij i,j≥1 ij i,j≥1 (finite or infinite). The Schur product of A and B is defined to be the matrix of elementwise products A∗B = [a b ] . ij ij i,j≥1 For M ∈ B(ℓ ) we let S denote the Schur multiplication map by 2 M M on B(ℓ ), S (A) = M ∗A. Then M is said to be a Schur multiplier 2 M if kS : B(ℓ ) → B(ℓ )k < +∞. M 2 2 If M = [m ] is a Schur multiplier and the iterated row and column ij limits lim(lim m ) = C and lim(lim m ) = R ij ij j→∞ i→∞ i→∞ j→∞ exist, then [Be] C = R. This shows in particular that j −i i+j +1 (cid:20) (cid:21)i,j≥1 is not a Schur multiplier. Theorem 3.1. Let (a ) be a sequence of reals which converges to n n≥1 0. Set b = ∆1(a ) = a −a and c = ∆2(a ) = a −2a +a . n n n n+1 n n n n+1 n+2 If the sequences (a /n), (b ) and (nc ) are all absolutely summable, n n n then the matrix (j −i)a i+j i+j +1 (cid:20) (cid:21)i,j≥1 is a Schur multiplier. 8 C. BADEA ANDV.I. PAULSEN Proof. The proof will be based on the following criterion due to Ben- nett [Be, Theorem 8.6] : if M = [m ] satisfies ij i,j≥1 limm = limm = 0 ij ij i j and +∞ (3.1) |m −m −m +m | < +∞, i,j i,j+1 i+1,j i+1,j+1 i,j=1 X then M is a Schur multiplier. Let m = (j −i)a /(i+j +1). We have ij i+j limm = limm = 0. ij ij i j In order to prove (3.1), we write ∞ |m −m −m +m | i,j i,j+1 i+1,j i+1,j+1 i,j=1 X ∞ (j −i)a 2(j −i)a (j −i)a i+j i+j+1 i+j+2 = − + i+j +1 i+j +2 i+j +3 n=1i+j=n(cid:12) (cid:12) X X (cid:12) (cid:12) (cid:12)∞ (cid:12) a 2a a (cid:12) n n+1 n+2 (cid:12) ≤ n(n−1) − + n+1 n+2 n+3 n=1 (cid:12) (cid:12) X (cid:12) (cid:12) ∞ c (cid:12) 1 a a (cid:12) = n(n−1) n(cid:12) + n − n(cid:12)+2 n+2 n+2 n+1 n+3 n=1 (cid:12) (cid:20) (cid:21)(cid:12) X (cid:12) (cid:12) ∞ c (cid:12) b +b 2a (cid:12) = n(n−1) n +(cid:12) n n+1 + n+2(cid:12) n+2 (n+1)(n+2) (n+1)(n+2)(n+3) n=1 (cid:12) (cid:12) X (cid:12) (cid:12) (cid:12) |a | (cid:12) n+2 ≤ n(cid:12)|c |+ |b |+ |b |+2 . (cid:12) n n n+1 n+2 n≥1 n≥1 n≥1 n≥1 X X X X All four sums are convergent by hypothesis. Corollary 3.2. Let ε > 0. The matrices j −i E = ε (i+j +1)(log(i+j +1))1+ε (cid:20) (cid:21)i,j≥1 and j −i F = ε (i+j +1)(log(i+j +1))(loglog(i+j +1))1+ε (cid:20) (cid:21)i,j≥1 FOGUEL-HANKEL OPERATORS 9 are Schur multipliers. Proof. Theorem 3.1 applies for the sequences 1 a = ,n ≥ 2 n (logn)1+ε and 1 (3.2) a = ,n ≥ 2. n (logn)[log(logn)]1+ε Wegivetheproofonlyforthesecondsequence(3.2). Denoter = 1+ε and log (x) = log(logx). The series 2 |a | 1 n = n n(logn)[log (n)]r n≥2 n≥2 2 X X is convergent since r > 1. Consider the C1 function f(x) = (logx)[log (x)]r 2 with (log (x))r +r(log (x))r−1 f′(x) = 2 2 . x We have f(n+1)−f(n) b = . n log(n)[log (n)]rlog(n+1)[log (n+1)]r 2 2 For each n there is a point θ between 0 and 1 such that n f(n+1)−f(n) = f′(n+θ ). n We obtain (log (n+θ ))r +r(log (n+θ ))r−1 |b | = 2 n 2 n n (n+θ )log(n)[log (n)]rlog(n+1)[log (n+1)]r (cid:12) n 2 2 (cid:12) (cid:12) (cid:12) and thus (cid:12) (cid:12) (cid:12) 1+r (cid:12) |b | ≤ n n(logn)[log (n)]r 2 for sufficiently large n. Thus the sequence (b ) is absolutely summable. n Consider now the C2 function g(x) = 1/f(x), with its second deriv- ative given by 1 g′′(x) = x2(logx)2(log x)r 2 10 C. BADEA ANDV.I. PAULSEN 2 3r + + x2(logx)3(log x)r x2(logx)3(log x)1+r 2 2 r r(1+r) + + . x2(logx)2(log x)1+r x2(logx)3(log x)2+r 2 2 For each n, there is η between 0 and 2 such that n c = g(n)−2g(n+1)+g(n+2) = g′′(n+η ). n n Using this representation of c it can be proved that (nc ) is absolutely n n summable. 4. A sufficient condition for similarity of R(X) to R(0) The idea of the proof of the following theorem goes back to [FW] and [W]. Theorem 4.1. Let T ∈ B(H ) be an isometry (T∗T = I ) and let 2 2 2 2 H2 T ∈ B(H ) and X : H → H be bounded operators. Consider 1 1 1 2 T∗ X R(T∗,T ;X) = 2 . 2 1 0 T " 1 # If n−1 (4.1) supk Tj+1XTjk < +∞, 2 1 n≥1 j=0 X then R(T∗,T ;X) is similar to T∗ ⊕T = R(T∗,T ;0). 2 1 2 1 2 1 Proof. Let LbeaBanachlimit [C], thatisaboundedlinear functional on ℓ (C) such that 1 = L(1) = kLk and L((x ) ) = L((x ) ) ∞ n+1 n≥0 n n≥0 for every (x ) ∈ ℓ (C). Here 1 = (1,1,...). n n≥0 ∞ Consider the linear operator Z : H → H given by 1 2 n−1 hZh ,h i = L h Tj+1XTjh ,h i . 1 2 H2 2 1 1 2 ! j=0 X Then (4.1) shows that Z is well-defined and bounded.