A family of Schur multipliers for lower triangular matrices with applications
aa r X i v : . [ m a t h . C A ] M a y A FAMILY OF SCHUR MULTIPLIERS FOR TRIANGULAROPERATORS WITH APPLICATIONS
N. CHALMOUKIS AND G. STYLOGIANNISA
BSTRACT . We construct a family of Schur multipliers for lower trian-gular matrices on 𝓁 𝑝 , < 𝑝 < ∞ related to 𝜃 -summability kernels, aclass of kernels including the classical Fejer, Riesz and Bochner kernels.From this simple fact we derive diverse applications. Firstly we find anew class of Schur multipliers for Hankel operators on 𝓁 , generalizinga result of E. Ricard. Secondly we prove that any space of analytic func-tions in the unit disc which can be identified with a weighted 𝓁 𝑝 space,has the property that the space of its multipliers is contained in the spaceof symbols 𝑔 that induce a bounded generalized Cesáro operator 𝑇 𝑔 .
1. I
NTRODUCTION
Consider the classical sequence spaces 𝓁 𝑝 , < 𝑝 < ∞ of all 𝑝 − summableinfinite sequences { 𝑥 𝑘 } 𝑘 ≥ . These spaces have a “privileged" Schauder ba-sis, namely the elements 𝑒 𝑘 ∶= { 𝛿 𝑘𝑛 } 𝑘 ≥ , with respect to which a boundedlinear operator 𝐴 ∈ ( 𝓁 𝑝 ) has a representation [ 𝐴 ] = ( ⟨ 𝐴𝑒 𝑘 , 𝑒 𝑛 ⟩ ) ≤ 𝑘,𝑛 as an infinite matrix, where the pairing ⟨ ⋅ , ⋅ ⟩ is the standard 𝓁 𝑝 − pairing.Usually we denote by 𝑎 𝑘𝑛 the entries of the matrix representation. Fromnow on we shall make little distinction between a bounded linear operatoritself and its matrix representation. A Schur multiplier is an infinite matrix 𝑆 = ( 𝜎 𝑘𝑛 ) 𝑘𝑛 such that 𝑆 ⊙ 𝐴 ∶= ( 𝜎 𝑘𝑛 𝑎 𝑘𝑛 ) 𝑘𝑛 ∈ ( 𝓁 𝑝 ) , ∀ 𝐴 ∈ ( 𝓁 𝑝 ) . The pointwise product ⊙ is usually referred to as Schur or Hadamard mul-tiplication. An application of the closed graph theorem shows, that we cannaturally define a norm on the space of Schur multipliers ‖ 𝑆 ‖ 𝑝 ∶= sup{ ‖ 𝑆 ⊙ 𝐴 ‖ ( 𝓁 𝑝 ) ∶ ‖ 𝐴 ‖ ( 𝓁 𝑝 ) ≤ . With this norm and the ⊙ multiplication the space of all Schur multipliers,denoted by 𝑝 is a commutative Banach algebra. The study of this space, as Mathematics Subject Classification.
Key words and phrases.
Schur multipliers, Hankel operators, Generalized Cesáro oper-ator, Fejer-Riesz-Bochner kernels.The research of the first author is supported by the fellowship INDAM-DP-COFUND-2015 "INdAM Doctoral Programme in Mathematics and/or Applications cofund by MarieSklodowska-Curie Actions" an object with inherent interest, has been initiated in the seminal paper ofBennett [4]. Since then, there has been a growing interest in Schur multipli-ers (see for example [1], [5], [12] ).In particular it seems the case that quite a few problems in operator theory,but also in the study of spaces of analytic functions can be formulated interms of Schur multipliers. We will return to this point later, providing alsosome applications of our main results.Often in applications one only needs to know that a given matrix is a Schurmultiplier for a sub-class of matrices, more often than not for Hankel andToeplitz matrices. The drawback when studying Schur multipliers for suchclasses is that a product of two Schur multipliers for e.g. Hankel matricesneed not be a Schur multiplier for Hankel matrices (same for Toeplitz etc)hence such Schur multipliers do not form a Banach algebra.What we propose here is to study Schur multipliers for lower (equiva-lently upper) triangular operators. By lower triangular operator we intent anoperator which leaves invariant the set of functions which a have a zero oforder 𝑛 at the origin, for every 𝑛 ≥ .This choice is motivated by several reasons. For one thing, Schur mul-tipliers for lower triangular operators form a closed Banach sub-algebra of 𝑝 , which we shall denote by ◺ 𝑝 , and therefore the tools from the classicaltheory of Banach algebras are available. Moreover, a little less obvious isthat elements in ◺ are in fact Schur multipliers for Hankel operators on 𝓁 (see Section 2.1).For Toeplitz operators this is not the case, but we will see that many ex-amples of matrices in ◺ that we are going to construct will in fact be alsoSchur multipliers for Toeplitz matrices.Our first main theorem provides a simple way of constructing non trivialSchur multipliers for lower triangular matrices. Theorem 1.1.
Let 𝜃 ∈ 𝐶 ( ℝ ) with support in [−1 , , such that | ̂𝜃 ( 𝑥 ) | = ( | 𝑥 | − 𝑎 ) , for some 𝑎 > . Then the matrix Θ ∶= { 𝜃 ( 𝑘𝑛 + 1 )} ≤ 𝑘 ≤ 𝑛 is a Schur multiplier for lower triangular matrices on 𝓁 𝑝 , < 𝑝 < ∞ . The proof of the above theorem uses only elementary techniques. In factif one considers the kernel 𝐾 𝜃𝑛 corresponding to the generating function 𝜃𝐾 𝜃𝑛 ( 𝑡 ) ∶= ∑ | 𝑘 | ≤ 𝑛 𝜃 ( 𝑘𝑛 + 1 ) 𝑒 𝑖𝑘𝑡 , the theorem follows by elementary manipulations and some simple point-wise estimates of these kernels.There is a variety of examples of functions 𝜃 satisfying the hypothesis ofthe theorem. A comprehensive list for example can be found in [21, Chapter FAMILY OF SCHUR MULTIPLIERS... 3
Corollary 1.2.
Let 𝛾 > , 𝑎 > , 𝑝 > , we define the matrix 𝐹 𝑎,𝛾 ∶= (( ( 𝑘𝑛 + 1 ) 𝛾 ) 𝑎 ) ≤ 𝑘 ≤ 𝑛 . Then 𝐹 𝑎,𝛾 belongs to ◺ 𝑝 . Moreover we have the Schur norm estimates(1) ‖ 𝐹 𝑎,𝛾 ‖ 𝑆 ◺ 𝑝 = 𝛾 ( 𝑎 ) , for ℕ ∋ 𝑎 → ∞ , 𝛾 > ,(2) ‖ 𝐹 ,𝛾 ‖ 𝑆 ◺ 𝑝 = 𝑎 ( 𝛾 ) , for 𝑎, 𝛾 ∈ ℕ ⧵ {0} . The big O notation should be interpreted as usual. For example in (1) itmeans that there exists a positive constant 𝑐 𝛾 depending on 𝛾 such that for 𝑎 sufficiently large natural number, the quantity 𝑐 𝛾 𝑎 bounds the norm of 𝐹 𝑎,𝛾 and similarly for (2). This quantitative estimates are important for furtherapplications of the theorem and they are obtained by a careful examinationof the constants involved (see Section 3).It is interesting to notice that in the special case 𝛾 = 𝑎 = 1 we have thefamiliar coefficients of the Fejer kernel. A natural question that comes tomind is how sharp is such a theorem. The case of the Fejer kernel is quiteindicative. First of all 𝐹 , is not a Schur multiplier on 𝓁 𝑝 for that would defythe following criterion by Bennett [4] for 𝑝 = 2 and Coine [11] for 𝑝 ≠ .Suppose that 𝑆 = ( 𝜎 𝑘,𝑛 ) is a Schur multiplier and that the iterated limits lim 𝑘 lim 𝑛 𝜎 𝑘,𝑛 =∶ 𝓁 , lim 𝑛 lim 𝑘 𝜎 𝑘𝑛 =∶ 𝓁 exist. Then 𝓁 = 𝓁 . An easy way to get around this problem is to ask if thematrix ( 𝑘𝑛 + 1 ) ≤ 𝑘 ≤ 𝑛 is a Schur multiplier, which differs by 𝐹 , only by the identity in 𝑆 ◺ 𝑝 . Sur-prisingly enough, even this matrix satisfies Bennett’s criterion it is still not aSchur multiplier. This can be seen by considering it’s action on the discreteHilbert transform. ∶= ( 𝑛 − 𝑘 ) 𝑘 ≠ 𝑛 . One has, ( 𝑘𝑛 + 1 ) ≤ 𝑘<𝑛 ⊙ = ( 𝑛 − 𝑘 + 1 𝑛 − 𝑘 𝑛 + 1 ) ≤ 𝑘<𝑛 + ( 𝑛 − 𝑘 ) ≤ 𝑘<𝑛 . One the right we have the sum of a bounded operator (dominated by twice theCesáro matrix) and an unbounded one (the lower truncation of the discreteHilbert transform), which proves our point. Similar considerations can becarried out for other values of 𝛾 and 𝑎. Our second result is more or less a consequence of Corollary 1.2 and theelementary theory of Banach algebras. Again we shall see in Section 2.1that this theorem has interesting applications.
N. CHALMOUKIS AND G. STYLOGIANNIS
Theorem 1.3.
For any 𝜆 ∉ 𝐷 (0 ,
1) ∩ 𝐷 (−1 , , 𝛾 > , the matrices 𝑋 ∶= { ( 𝑛 + 1) 𝛾 𝑘 𝛾 + 𝜆 ( 𝑛 + 1) 𝛾 } ≤ 𝑘 ≤ 𝑛 , 𝑋 ∶= { 𝑘 𝛾 𝑘 𝛾 + 𝜆 ( 𝑛 + 1) 𝛾 } ≤ 𝑘 ≤ 𝑛 belong to ◺ 𝑝 , < 𝑝 < ∞ . Organization of the material.
We have organized the paper as follows. Inthe next section we shall discuss the applications of our main results givingthe proofs assuming that Theorems 1.1 and 1.3 are true. In Section 3 weprove Theorems 1.1 and 1.3. In the last section we discuss a mostly openproblem about the extensibility of an element in 𝓁 𝑝 . Finally there existsan appendix wherein we provide the pointwise estimates of the Fejer-Rieszkernels with appropriate constants.2. A PPLICATIONS
Schur multipliers for Hankel matrices.
The first application we aregoing to draw is a generalization of the following result due to Ricard [20].
Theorem 2.1 ([20]) . The matrix ( 𝑘 + 1 𝑘 + 𝑛 + 1 ) 𝑘,𝑛 is a Schur multiplier for bounded Hankel matrices on 𝓁 . It worth’s mentioning that this result apart from the independent interestthat it might have, it answers a question of Davindson and Paulsen aboutCAR-valued Foguel-Hankel operators which are similar to a contraction (formore details see [20] and [13]). The following corollary can be consideredas an asymmetric version of Ricard’s theorem.
Corollary 2.2.
For Re 𝜆 > , the matrices ( 𝑘 + 1 𝑘 + 𝜆𝑛 + 1 ) 𝑘,𝑛 are Schur multipliers for Hankel operators on 𝓁 . Proof.
This is more or less a corollary of Theorem 1.3 together with a the-orem of Bonami and Bruna in [6], which says that if is a bounded Han-kel operator on 𝓁 then its triangular truncation, denoted by Π( ) is alsobounded on 𝓁 . Therefore if 𝑆 ∈ ◺ 𝑆 ⊙ = 𝑆 ⊙ Π( ) ∈ ( 𝓁 ) . Now notice that since Re 𝜆 > it holds true that 𝜆, 𝜆 ∉ 𝐷 (−1 , , there-fore Theorem 1.3 applies to both 𝜆 and its reciprocal. Furthermore,(1) 𝑘 + 1 𝑘 + 𝜆𝑛 + 1 − 𝑘 + 1 𝑘 + 𝜆𝑛 + 𝜆 = ( 𝑘 + 1)( 𝜆 − 1)( 𝜆𝑛 + 𝑘 + 1)( 𝜆𝑛 + 𝑘 + 𝜆 ) . FAMILY OF SCHUR MULTIPLIERS... 5
Which proves that also the matrix ( 𝑘 + 1 𝑘 + 𝜆𝑛 + 1 ) 𝑘 ≤ 𝑛 is a Schur multiplier for lower triangular matrices, since the matrix on theright hand side of equation (1) is bounded as a product of a bounded Hilberttype matrix and a matrix in 𝑆 ◺ , by Theorem 1.3. Since also the set of Schurmultipliers for Hankel operators is closed under taking the transpose of amatrix ( 𝑘 + 1 𝑘 + 𝜆𝑛 + 1 ) 𝑘,𝑛 = ( 𝑘 + 1 𝑘 + 𝜆𝑛 + 1 ) 𝑘 ≤ 𝑛 + 1 𝜆 ( 𝑛 + 1 𝑘 + 𝜆 ( 𝑛 + 1) ) 𝑡𝑘 ≤ 𝑛 − diag { 𝑛 + 1 𝜆 ( 𝑛 + 1) + 1 } 𝑛 ≥ . Which justifies the claim. (cid:3)
It would be interesting to know if the theorem of Bonami and Bruna re-mains valid when 𝑝 ≠ , or for weighted 𝓁 spaces, but this appears to bea subtle question. The problem seems to be that the proof of Bonami andBruna uses the Nehari theorem and also some delicate estimates for the bi-linear Hilbert transform due to Lacey and Thiele [16].We should also mention that our approach is completely different fromthe one followed by Ricard, which uses Hardy space theory.2.2. Generalized Cesáro operators on weighted 𝓁 𝑝 spaces. If one iden-tifies in the usual way the Hardy space in the disc 𝐻 ( 𝔻 ) of power serieswith square summable coefficients, with 𝓁 , then the Cesáro operator takesthe form of a weighted integration operator, which we call 𝐶 , 𝐶𝑓 ( 𝑧 ) = 1 𝑧 ∫ 𝑧 𝑓 ( 𝑡 )1 − 𝑡 𝑑𝑡. There is a certain generalization of this operator which turns out to be impor-tant for applications in Hardy space theory. Namely for an analytic function 𝑔 in the disc we define the generalized Cesáro operator 𝑇 𝑔 𝑓 ( 𝑧 ) ∶= ∫ 𝑧 𝑓 ( 𝑡 ) 𝑔 ′ ( 𝑡 ) 𝑑𝑡. (The factor 𝑧 −1 is ommited since it doesn’t change the boundedness prop-erties of 𝑇 𝑔 ). Such operators, originated in the work of Calderon [9] andPommerenke [19] have been studied in most of the prominent examples ofspaces of analytic functions (see for example [2], [3], [7], [14], [17]).Here we would like to focus on a particular feature of these operators,which connects them with the multiplier space of a space of analytic func-tions.Let 𝑋 a Banach space of analytic functions in the unit disc and supposeagain 𝑔 is an analytic symbol. We can define the multiplication operator as N. CHALMOUKIS AND G. STYLOGIANNIS follows 𝑀 𝑔 𝑓 ∶= 𝑓 𝑔, ∀ 𝑓 ∈ 𝑋. The space of symbols 𝑔 such that the corresponding operator 𝑀 𝑔 is boundedon 𝑋 , equipped with the norm induced by the operator 𝑀 𝑔 is called the mul-tiplier space of 𝑋 and denoted usually by Mult( 𝑋 ) . Same kind of reason-ing leads to the space 𝑇 ( 𝑋 ) , the space of symbols that induced generalizedCesáro operators which are bounded on 𝑋 . Both spaces are in all concretecases very important spaces associated to the function theory in the space 𝑋 .For example if 𝑋 = 𝐻 𝑝 , the Hardy space in the unit disc, then Mult( 𝐻 𝑝 ) = 𝐻 ∞ and 𝑇 ( 𝐻 𝑝 ) = 𝐵𝑀 𝑂𝐴 (analytic functions of bounded mean oscillation), when ≤ 𝑝 < ∞ , or if 𝑋 = 𝐴 𝑝 , ≤ 𝑝 < ∞ , the Bergman space in theunit disc, then Mult( 𝐴 𝑝 ) = 𝐻 ∞ , 𝑇 ( 𝐴 𝑝 ) = (Bloch space).The common pattern seems to be that, even if 𝑀 ( 𝑋 ) contains only boundedanalytic functions the space 𝑇 ( 𝑋 ) is a space strictly containing Mult( 𝑋 ) ,which is more adopted to the function theory in 𝑋 . This folklore percep-tion is often expressed by saying that 𝐵𝑀 𝑂𝐴 is the “right analogue" of 𝐻 𝑝 when 𝑝 → ∞ , and similarly for and 𝐴 𝑝 .The fact that Mult( 𝑋 ) ⊆ 𝑇 ( 𝑋 ) can be intuitively explained by the follow-ing observation. Suppose that we call 𝜕 the derivative operator, and I theintegration operator, i.e. I 𝑓 ( 𝑧 ) ∶= ∫ 𝑧 𝑓 ( 𝑡 ) 𝑑𝑡, we have a relation of the form 𝑀 𝑔 − 𝑇 𝑔 = I ◦ 𝑀 𝑔 ◦ 𝜕. Although this suggest a similarity between the operators 𝑀 𝑔 and 𝑀 𝑔 − 𝑇 𝑔 ,it has to be taken with a grain of salt. The problem is that 𝜕 usually is notbounded on 𝑋 and I is not exactly the inverse of 𝜕 .When, instead, one asks even the most basic questions about Mult( 𝓁 𝑝 ) and 𝑇 ( 𝓁 𝑝 ) , 𝑝 ≠ they seem intractable. More precisely we identify the se-quences in 𝓁 𝑝 with the space of analytic functions in the unit disc with thesame Taylor coefficients, therefore we can see 𝓁 𝑝 as a Banach space of an-alytic functions in the unit disc and the above definitions make sense. Theproblem of understanding Mult( 𝓁 𝑝 ) is not new (see the interesting survey[10]) with a lot of open problems. While there hasn’t been systematic re-search on 𝑇 ( 𝓁 𝑝 ) (the authors intent to return to the study of this space in afeature work).But what about our previous remark? Is it true at least that Mult( 𝓁 𝑝 ) ⊆𝑇 ( 𝓁 𝑝 ) , < 𝑝 < ∞? We are able to give an affirmative answer in great gen-erality using again the Schur multipliers for lower triangular matrices thatwe constructed. The class of spaces in which our result applies are spacesof analytic functions in the unit disc which can be isometrically identifiedwith a weighted 𝓁 𝑝 space. To be more precise, let 𝜔 ∶= { 𝜔 𝑛 } a sequence of FAMILY OF SCHUR MULTIPLIERS... 7 (positive) weights, such that(2) lim sup 𝑛 𝜔 𝑛 𝜔 𝑛 +1 ≤ . Then for a sequence { 𝑎 𝑛 } ∈ 𝓁 𝑝 ( 𝜔 ) the powerseries with coefficients { 𝑎 𝑛 } converges in the unit disc and therefore it has meaning to talk about 𝓁 𝑝 ( 𝜔 ) as a space of analytic functions in the unit disc. Theorem 2.3.
For any 𝑝 ∈ (1 , ∞) , and any weight 𝜔 which satisfies (2) Mult( 𝓁 𝑝 ( 𝜔 )) ⊆ 𝑇 ( 𝓁 𝑝 ( 𝜔 )) . Proof.
By the definition of the norms involved, the diagonal operator 𝑈 ∶ 𝓁 𝑝 ↦ 𝓁 𝑝 ( 𝜔 ) 𝑈 ∶= diag { 𝜔 𝑝𝑛 } is a surjective isometry. Therefore by a standard computation on the mono-mials we see that for an analytic symbol 𝑔 with coefficients 𝑔 𝑘 the multipli-cation operator 𝑀 𝑔 on 𝓁 𝑝 ( 𝜔 ) is isometrically equivalent (through 𝑈 ) to theoperator ( 𝑔 𝑛 − 𝑘 𝜔 𝑝𝑛 𝜔 𝑝𝑘 ) ≤ 𝑘 ≤ 𝑛 , acting on the (unweighted) 𝓁 𝑝 . While 𝑇 𝑔 is isometrically equivalent to (( 𝑘𝑛 + 1 ) 𝑔 𝑛 − 𝑘 𝜔 𝑝𝑛 𝜔 𝑝𝑘 ) ≤ 𝑘 ≤ 𝑛 , on 𝓁 𝑝 . At this point the theorem is a direct corollary of Theorem 1.1 for theFejer kernel. (cid:3)
An interesting consequence of this theorem is that it provides a completelyoperator theoretic proof of the fact that 𝐻 ∞ ⊆ 𝐵𝑀 𝑂𝐴, .
3. P
ROOF OF THE MAIN THEOREMS
The proof of the the following elementary lemma can be found scatteredaround the literature. We provide a short proof of it for the sole purpose ofcompleteness.
Lemma 3.1.
Let 𝜃 ∈ 𝐶 ( ℝ ) with support contained in [−1 , , and | ̂𝜃 ( 𝑥 ) | = ( | 𝑥 | − 𝑎 ) , for some 𝑎 > . Then the kernel 𝑘 𝜃𝑛 ( 𝑡 ) ∶= ∑ | 𝑘 | ≤ 𝑛 𝜃 ( 𝑘𝑛 + 1 ) 𝑒 𝑖𝑘𝑡 satisfies(1) ‖ 𝑘 𝜃𝑛 ‖ 𝐿 ( 𝕋 ) ≤ ‖ ̂𝜃 ‖ 𝐿 ( ℝ ) . N. CHALMOUKIS AND G. STYLOGIANNIS (2) | 𝑘 𝜃𝑛 ( 𝑥 ) | ≤ 𝑐 min{ 𝑛 + 1 , ( 𝑛 + 1) −( 𝑎 −1) | 𝑥 | − 𝑎 } , for an absolute constant 𝑐 > . Proof.
Consider the corresponding continuous kernel 𝐾 𝜃𝑇 ( 𝑥 ) ∶ = ∫ 𝑇 − 𝑇 𝜃 ( 𝑡𝑇 ) 𝑒 𝑖𝑡𝑥 𝑑𝑡 = 𝑇 ̂𝜃 ( 𝑥𝑇 ) . By the Poisson summation formula we have that 𝑘 𝜃𝑛 ( 𝑥 ) = ∑ 𝑘 ∈ ℤ 𝐾 𝜃𝑛 ( 𝑥 + 2 𝑘𝜋 ) . Hence, ‖ 𝑘 𝜃𝑛 ‖ 𝐿 ( 𝕋 ) ≤ ‖ 𝐾 𝜃𝑛 ‖ = ‖ ̂𝜃 ‖ 𝐿 ( ℝ ) < +∞ . Also, | 𝑘 𝜃𝑛 ( 𝑥 ) | ≤ (2 𝑛 + 1) ‖ 𝜃 ‖ 𝐿 ∞ ( ℝ ) . Finally, | 𝑘 𝜃𝑛 ( 𝑥 ) | ≤ 𝑛 ∑ 𝑘 ∈ ℤ | ̂𝜃 ( 𝑛𝑥 + 2 𝑘𝑛𝜋 ) | ≤ 𝑐𝑛 ∑ 𝑘 ∈ ℤ | 𝑥𝑛 + 2 𝑘𝜋𝑛 | 𝑎 ≤ 𝑐 𝑎𝑎 − 1 1 | 𝑥 | 𝑎 𝑛 𝑎 −1 , 𝑥 ≠ . (cid:3) Theorem 1.2 will now follow from a slightly more general result.
Theorem 3.2.
Let { 𝜑 𝑛 } ⊂ 𝐿 ( 𝕋 ) be a family of kernels which satisfy: (1) 𝜑 𝑛 is a trigonometric polynomial of degree 𝑛 . (2) ‖ 𝜑 𝑛 ‖ 𝐿 ≤ 𝜌 , (3) | 𝜑 𝑛 ( 𝑡 ) | ≤ 𝜌 min{ 𝑛 + 1 , 𝑛 +1) 𝑎 𝑡 𝑎 +1 } , 𝑎 > , − 𝜋 < 𝑡 < 𝜋. Then the matrix
Φ ∶= { ̂𝜑 𝑛 ( 𝑘 ) } 𝑘,𝑛 is a Schur multiplier for lower triangular matrices on 𝓁 𝑝 for 𝑝 > . Further-more, ‖ Φ ‖ ◺ 𝑝 = 𝑎,𝑝 ( 𝜌 ) , as 𝜌 → ∞ . Proof.
Suppose now that 𝑥 = ( 𝑥 𝑘 ) ∈ 𝓁 𝑝 . By Hölder’s inequality and thefact that ‖ 𝜑 𝑛 ‖ 𝐿 ≤ 𝜌 we have that, FAMILY OF SCHUR MULTIPLIERS... 9 ‖ Φ ⊙ 𝐴 ( 𝑥 ) ‖ 𝑝 𝓁 𝑝 = ∞ ∑ 𝑛 =0 ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 ̂𝜑 𝑛 ( 𝑘 ) 𝑥 𝑘 ||||| 𝑝 = ∞ ∑ 𝑛 =0 ||||| ∫ 𝜋 − 𝜋 𝜑 𝑛 ( 𝑡 ) 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 𝑒 𝑖𝑘𝑡 𝑑𝑡 𝜋 ||||| 𝑝 ≤ 𝜌 𝑝𝑞 ∞ ∑ 𝑛 =0 ∫ 𝜋 − 𝜋 || 𝜑 𝑛 ( 𝑡 ) || ⋅ ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 𝑒 𝑖𝑘𝑡 ||||| 𝑝 𝑑𝑡 𝜋 ≤ 𝑝 −1 𝜌 𝑝𝑞 ∞ ∑ 𝑛 =0 ∫ 𝜋 − 𝜋 || 𝜑 𝑛 ( 𝑡 ) || ⋅ ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) ||||| 𝑝 𝑑𝑡 𝜋 + 2 𝑝 −1 𝜌 𝑝𝑞 ∞ ∑ 𝑛 =0 ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ||||| 𝑝 ≤ 𝑝 −1 𝜌 𝑝 ∞ ∑ 𝑛 =0 ( 𝑛 + 1) ∫ | 𝑡 | < 𝑛 +1 ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) ||||| 𝑝 𝑑𝑡 𝜋 + 2 𝑝 −1 𝜌 𝑝 ∞ ∑ 𝑛 =0 ∫ 𝑛 +1 < | 𝑡 | <𝜋 𝑛 + 1) 𝑎 | 𝑡 | 𝑎 +1 ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) ||||| 𝑝 𝑑𝑡 𝜋 + 2 𝑝 −1 𝜌 𝑝𝑞 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 . Lets call the two main terms appearing above (I) and (II) in order of appear-ance. In order to estimate these terms, for 𝑛 ∈ ℕ we define(3) 𝑆 𝑛 ( 𝑡 ) ∶= 𝑛 ∑ 𝜆 =0 |||||| 𝜆 ∑ 𝑘 =0 𝑎 𝜆𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) |||||| 𝑝 . Notice that if we define a sequence 𝑦 𝑘 = 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) for ≤ 𝑘 ≤ 𝑛 , 𝑦 𝑘 = 0 otherwise we have that(4) 𝑆 𝑛 ( 𝑡 ) = 𝑛 ∑ 𝜆 =0 |||||| 𝜆 ∑ 𝑘 =0 𝑎 𝜆𝑘 𝑦 𝑘 |||||| 𝑝 ≤ ‖ 𝐴 ( 𝑦 ) ‖ 𝑝 𝓁 𝑝 ≤ ‖ 𝐴 ‖ 𝑝 𝑛 ∑ 𝑘 =0 | 𝑥 𝑘 | 𝑝 | 𝑒 𝑖𝑘𝑡 − 1 | 𝑝 . (Notice that this is the only place where we use the assumption that 𝐴 islower triangular.) With this estimate in hand we go back to estimate (I) and (II). Fix 𝑀 > and by Abel’s summation by parts we haveI = ∫ 𝜋 − 𝜋 𝑀 ∑ 𝑛 =0 ( 𝑛 + 1) 𝜒 [ | 𝑡 | < 𝑛 +1 ] ( 𝑡 ) ||||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) ||||| 𝑝 𝑑𝑡 𝜋 = ∫ 𝜋 − 𝜋 𝑀 ∑ 𝑛 =0 ( 𝑛 + 1) 𝜒 [ | 𝑡 | < 𝑛 +1 ] ( 𝑡 ) ( 𝑆 𝑛 ( 𝑡 ) − 𝑆 𝑛 −1 ( 𝑡 )) ) 𝑑𝑡 𝜋 = ∫ 𝜋 − 𝜋 𝑀 −1 ∑ 𝑛 =0 ( ( 𝑛 + 1) 𝜒 [ | 𝑡 | < 𝑛 +1 ] ( 𝑡 ) − ( 𝑛 + 2) 𝜒 [ | 𝑡 | < 𝑛 +2 ] ( 𝑡 ) ) 𝑆 𝑛 ( 𝑡 ) 𝑑𝑡 𝜋 + ∫ 𝜋 − 𝜋 ( 𝑀 + 1) 𝜒 [ | 𝑡 | < 𝑀 +1 ] ( 𝑡 ) 𝑆 𝑀 ( 𝑡 ) 𝑑𝑡 𝜋 ≤ 𝑀 −1 ∑ 𝑛 =0 ( 𝑛 + 1) ∫ 𝑛 +2 < | 𝑡 | < 𝑛 +1 𝑆 𝑛 ( 𝑡 ) 𝑑𝑡 𝜋 + ( 𝑀 + 1) ∫ 𝑀 +1 − 𝑀 +1 𝑆 𝑀 ( 𝑡 ) 𝑑𝑡 𝜋 ≤ ∞ ∑ 𝑛 =0 ( 𝑛 + 1) ∫ 𝑛 +2 < | 𝑡 | < 𝑛 +1 𝑆 𝑛 ( 𝑡 ) 𝑑𝑡 𝜋 + 2 𝑝 𝜋 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 =∶ I ′ + 2 𝑝 𝜋 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 . But, I ′ ≤ ‖ 𝐴 ‖ 𝑝 ∞ ∑ 𝑛 =0 ( 𝑛 + 1) ∫ 𝑛 +11 𝑛 +2 𝑛 ∑ 𝑘 =0 | 𝑥 𝑘 | 𝑝 ( 𝑘𝑡 ) 𝑝 𝑑𝑡 ≤ ‖ 𝐴 ‖ 𝑝 ∞ ∑ 𝑘 =0 𝑘 𝑝 | 𝑥 𝑘 | 𝑝 ∞ ∑ 𝑛 = 𝑘 𝑛 + 1) 𝑝 +1 ≤ ‖ 𝐴 ‖ 𝑝 ∞ ∑ 𝑘 =0 𝑘 𝑝 | 𝑥 𝑘 | 𝑝 ∫ ∞ 𝑘 𝑥 𝑝 +1 𝑑𝑥 ≤ 𝑝 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 . For (II) we use a similar method. Fix
𝑀 > , FAMILY OF SCHUR MULTIPLIERS... 11 𝑀 ∑ 𝑛 =0 ∫ 𝑛 +1 < | 𝑡 | <𝜋 𝑛 + 1) 𝑎 | 𝑡 | 𝑎 +1 ||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) ||| 𝑝 𝑑𝑡 𝜋 = ∫ 𝜋 − 𝜋 𝑀 ∑ 𝑛 =0 𝑛 + 1) 𝑎 𝜒 [ 𝑛 +1 < | 𝑡 | <𝜋 ] ( 𝑡 ) ||| 𝑛 ∑ 𝑘 =0 𝑎 𝑛𝑘 𝑥 𝑘 ( 𝑒 𝑖𝑘𝑡 − 1) ||| 𝑝 𝑑𝑡 𝜋 | 𝑡 | 𝑎 +1 ≤ ∞ ∑ 𝑛 =0 ( 𝑛 + 1) 𝑎 − 1( 𝑛 + 2) 𝑎 ) ∫ 𝑛 +1 < | 𝑡 | <𝜋 𝑆 𝑛 ( 𝑡 ) 𝑑𝑡 𝜋 | 𝑡 | 𝑎 +1 + 1( 𝑀 + 1) 𝑎 ∫ 𝑀 +1 < | 𝑡 | <𝜋 𝑆 𝑀 ( 𝑡 ) 𝑑𝑡 𝜋 | 𝑡 | 𝑎 +1 ≤ ∞ ∑ 𝑛 =0 ( 𝑛 + 1) 𝑎 − 1( 𝑛 + 2) 𝑎 ) ∫ 𝑛 +1 < | 𝑡 | <𝜋 𝑛 ∑ 𝑘 =0 | 𝑥 𝑘 | 𝑝 | 𝑒 𝑖𝑘𝑡 − 1 | 𝑝 𝑑𝑡 𝜋 | 𝑡 | 𝑎 +1 ‖ 𝐴 ‖ 𝑝 + 2 𝑝 𝑎𝜋 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 = ∞ ∑ 𝑘 =0 | 𝑥 𝑘 | 𝑝 ∞ ∑ 𝑛 = 𝑘 ( 𝑛 + 1) 𝑎 − 1( 𝑛 + 2) 𝑎 ) ∫ 𝑛 +1 < | 𝑡 | <𝜋 | 𝑒 𝑖𝑘𝑡 − 1 | 𝑝 𝜋 | 𝑡 | 𝑎 +1 𝑑𝑡 ‖ 𝐴 ‖ 𝑝 + 2 𝑝 𝜋𝑎 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 = II ′ + 2 𝑝 𝑎𝜋 ‖ 𝐴 ‖ 𝑝 ‖ 𝑥 ‖ 𝑝 𝓁 𝑝 . We estimate the integral in II’ as follows: ∫ 𝑛 < | 𝑡 | <𝜋 | 𝑒 𝑖𝑘𝑡 − 1 | 𝑝 | 𝑡 | 𝑎 +1 𝑑𝑡 𝜋 ≤ 𝑘 𝑝 ∫ 𝑘 𝑛 𝑡 𝑝 − 𝑎 −1 𝑑𝑡 𝜋 + 2 𝑝 +1 ∫ 𝜋 𝑘 𝑡 𝑎 +1 𝑑𝑡 𝜋 ≤ 𝜋 ( 1 𝑝 − 𝑎 + 2 𝑝 𝑎 ) 𝑘 𝑎 . Here note that we can always assume that 𝑎 ≤ , so 𝑝 > 𝑎. Consequently,II ′ ≤ 𝜋 ( 1 𝑝 − 𝑎 + 2 𝑝 𝑎 ) 𝑘 𝑎 ∞ ∑ 𝑛 = 𝑘 ( 𝑛 + 1) 𝑎 − 1( 𝑛 + 2) 𝑎 ) ≤ 𝜋 ( 1 𝑝 − 𝑎 + 2 𝑝 𝑎 ) . (cid:3) Corollary 1.2 now follows by known estimates on the Fourier transformof an elementary function.
Lemma 3.3. [8, Lemma 2] Let 𝑎, 𝛾 > and 𝜙 𝑎,𝛾 ( 𝑥 ) ∶= max{0 , (1 − | 𝑥 | 𝛾 )} 𝑎 . Then ̂𝜙 𝑎,𝛾 ( 𝑥 ) ≤ 𝑐 ( 𝑎, 𝛾 ) | 𝑥 | − min{1 ,𝛾,𝑎 }−1 In order to obtain the quantitative behaviour of the Schur multiplier normclaimed in Corollary 1.2 we need the following quantitative estimates forthe constant 𝑐 ( 𝑎, 𝛾 ) in the previous lemma(1) 𝑐 ( 𝑎, 𝛾 ) = 𝛾 ( 𝑎 ) , for ℕ ∋ 𝑎 → ∞ , 𝛾 > ,(2) 𝑐 ( 𝑎, 𝛾 ) = 𝑎 ( 𝛾 ) , for 𝑎, 𝛾 ∈ ℕ ⧵ {0} . The way to obtain these estimates is based on calculus arguments and theproof of [8, Lemma 2]. A discussion of the proof can be found in the ap-pendix.The second main theorem will follow from some considerations on thespectrum of the elements in ◺ 𝑝 . Proof of Theorem 1.3.
Let us denote by 𝑋 ⊙𝑁 the 𝑁 th Hadamard power ofa matrix (i.e. the matrix with every entry elevated to the power 𝑁 ). FromTheorem 1.2 we have that lim 𝑁 → ∞ ‖ 𝐹 ⊙𝑁 ,𝛾 ‖ 𝑁 ◺ 𝑝 = lim 𝑁 → ∞ ‖ 𝐹 𝑁,𝛾 ‖ 𝑁 ◺ 𝑝 ≤ lim 𝑁 → ∞ 𝑁 𝑁 = 1 . The existence of the limit is guaranteed by the fact that we are in a Banachalgebra. By the spectral radius formula we know that 𝜎 ( 𝐹 ,𝛾 ) ⊆ 𝐷 (0 , , where 𝜎 ( 𝐹 ,𝛾 ) denotes the spectrum of the Schur multiplier operator. Denotenow by ◺ the lower triangular matrix with all entries below or on the diag-onal equal to . By a similar argument and a second application of Theorem1.2 lim 𝑁 → ∞ ‖ ( ◺ − 𝐹 ,𝛾 ) ⊙𝑁 ‖ 𝑁 ◺ 𝑝 = lim 𝑁 → ∞ ‖ ◺ − 𝐹 ,𝑁𝛾 ‖ 𝑁 ◺ 𝑝 = lim 𝑁 → ∞ ‖ ◺ − 𝐹 , ⌊ 𝑁𝛾 ⌋ −1 ‖ 𝑁 ◺ 𝑝 ≤ lim 𝑁 → ∞ ( ⌊ 𝑁 𝛾 ⌋ − 1) 𝑁 = 1 . Combining both estimates we arrive at 𝜎 ( 𝐹 ,𝛾 ) ⊆ 𝐷 (0 ,
1) ∩ 𝐷 (1 , . Hence, if 𝜆 ∉ 𝐷 (−1 ,
1) ∩ 𝐷 (0 , the matrix ( 𝜆 + 1) ◺ − 𝐹 ,𝛾 = ( 𝜆 + ( 𝑘𝑛 + 1 ) 𝛾 ) ≤ 𝑘 ≤ 𝑛 is invertible in ◺ 𝑝 . But the inverse is obtained just by taking the algebraicinverse of the entries of the matrix pointwise, i.e. (( 𝜆 + 1) ◺ − 𝐹 ,𝛾 ) −1 = ( ( 𝑛 + 1) 𝛾 𝑘 𝛾 + 𝜆 ( 𝑛 + 1) 𝛾 ) ≤ 𝑘 ≤ 𝑛 . FAMILY OF SCHUR MULTIPLIERS... 13
To see why also 𝑋 is in ◺ 𝑝 compute ( ◺ − 𝐹 ,𝛾 )(( 𝜆 + 1) ◺ − 𝐹 ,𝛾 ) −1 = ( 𝑘 𝛾 𝑘 𝛾 + 𝜆 ( 𝑛 + 1) 𝛾 ) ≤ 𝑘 ≤ 𝑛 , which also belongs to ◺ 𝑝 . (cid:3) Extensibility of Schur multipliers for lower triangular operators.
In this short section we would like to draw attention to a problem connectedto multipliers in ◺ 𝑝 that we think that is very interesting. Suppose that alower triangular matrix 𝑆 has the following property. There exists a matrix 𝑇 ∈ 𝑝 such that Π( 𝑇 ) = 𝑆. If this happens we say that 𝑇 extends to a Schur multiplier in 𝑆 𝑝 . Of courseif 𝑇 is such it is also a Schur multiplier for lower triangular matrices on 𝓁 𝑝 .The converse it is not at all clear. Problem 3.4.
Is it true that every element in ◺ 𝑝 , < 𝑝 < ∞ can be extendedto a Schur multiplier in 𝑝 ?In particular we do not know whether the examples constructed in Corol-lary 1.2 and Theorem 1.3 satisfy this extensibility property. A necessarycondition that a Schur multiplier for lower triangular operatrs has to satisfyin order to be extensible is to be completely bounded as an operator on thespace of bounded lower triangular operators. That is because every Schurmultiplier in 𝓁 is completely bounded [18, Theorem 5.1].In the particular case that 𝑎 ∶= { 𝑎 𝑘 } 𝑘 ≥ and 𝑝 = 2 is a sequence of com-plex numbers and 𝑇 𝑎 is the corresponding (lower triangular) Toeplitz matrix,it can be seen without difficulty that this is indeedÎť the case. Proposition 3.5.
For 𝑎 and 𝑇 𝑎 as before, 𝑇 𝑎 ∈ 𝑆 ◺ if and only if 𝜏 ( 𝑧 ) ∶= ∑ 𝑘 ≥ 𝑎 𝑘 𝑧 𝑘 is a Cauchy transform of a finite (complex) Borel meausre on the unit circle.Proof. If 𝜏 is such, that means that there exists a finite Borel measure 𝜇 suchthat 𝑎 𝑘 are the positive Fourier coefficients of 𝜇 . Then by [4, Theorem 8.1] 𝑇 𝑎 extends to a Schur multiplier in 𝑆 . Suppose now that 𝑇 𝑎 ∈ 𝑆 ◺ . Let usdenote by ∗ the coefficient wise multiplication of two power series. Thenfor a bounded holomorphic function ℎ we have ‖ 𝑇 𝑎 ⊙ 𝑇 ℎ ‖ 𝓁 = ‖ 𝑇 𝜏 ∗ ℎ ‖ 𝓁 = ‖ 𝜏 ∗ ℎ ‖ 𝐻 ∞ < +∞ . In other words 𝜏 is a coefficient self-multiplier for 𝐻 ∞ , and this is equivalent[15, Theorem 10.1.2] to being a Cauchy transform of a finite Borel measure. (cid:3) A similar reasoning, with the aid of the theorem of Bonami and Bruna[6] shows that this is also the case when we consider multipliers of the form Π( ) where is a Hankel type matrix.4. A PPENDIX
Proof of Lemma 3.3.
We have that(5) ̂𝜙 𝑎,𝛾 ( 𝑥 ) = 2 ∫ (1 − 𝑡 𝛾 ) 𝑎 𝑐𝑜𝑠 ( 𝑥𝑡 ) 𝑑𝑡. The proof breaks down in the following two cases.Case I: Let 𝑎 ∈ ℕ , 𝛾 ≥ . In this case we integrate by parts twice in (5) andestimate in the obvious way.Case II: Let 𝑎 ∈ ℕ , < 𝛾 < . This is the more tricky case and we generallyfollow [8, Lemma 2]. In order to not overload the exposition with calcula-tions we try only to highlight the modifications of the original proof in orderto get the quadratic growth.We make use of the following decomposition of unity: 𝜓 ( 𝑥 ) + 𝜓 ( 𝑥 ) + 𝜓 ( 𝑥 ) = 1 , 𝑥 ∈ [0 , ,𝜓 𝑖 ∈ 𝐶 ∞ 𝑐 ( ℝ ) as in [8, Lemma 2]. Let 𝐶 𝜓 𝑖 = max{sup{ | 𝜓 ( 𝑛 ) 𝑖 ( 𝑥 ) | , 𝑥 ∈ ℝ } , 𝑛 = 0 , , . Then ̂𝜙 𝑎,𝛾 ( 𝑥 ) = 𝐼 ( 𝑥 ) + 𝐼 ( 𝑥 ) + 𝐼 ( 𝑥 ) with 𝐼 𝑖 ( 𝑥 ) = ∫ (1 − 𝑡 𝛾 ) 𝑎 𝜓 𝑖 ( 𝑡 ) 𝑒 𝑖𝑥𝑡 𝑑𝑡 . By integrating 𝐼 twice by parts onefinds that | 𝐼 ( 𝑥 ) | ≤ ( 𝑎 𝛾 ( 𝛿 ) 𝛾 −2 + 1 ) 𝐶 𝜓 ⋅ | 𝑥 | 𝑛 = 𝐶 ( 𝑎, 𝛾 ) 1 | 𝑥 | 𝑛 , 𝑛 = 1 , . The integral 𝐼 is estimated as in [8, Lemma 2]. This gives : | 𝐼 ( 𝑥 ) | ≤ 𝑎 𝛾 + 2 𝑎 + 2 𝛾 𝐶 𝜓 ⋅ | 𝑥 | 𝛾 = 𝐶 ( 𝑎, 𝛾 ) 1 | 𝑥 | 𝛾 . For 𝐼 we work as follows: Performing the change of variable 𝑦 = 1 − 𝑡 gives 𝐼 ( 𝑥 ) = ∫ (1 − (1 − 𝑦 ) 𝛾 ) 𝑎 𝜙 (1 − 𝑦 ) 𝑒 𝑖𝑥 (1− 𝑦 ) 𝑑𝑦. Integrating by parts twice we estimate: | 𝐼 ( 𝑥 ) | ≤ ( 𝑎 𝛾 (1 − 𝛿 ) 𝛾 −2 ) 𝐶 𝜓 ⋅ | 𝑥 | 𝑛 = 𝐶 ( 𝑎, 𝛾 ) 1 | 𝑥 | 𝑛 , 𝑛 = 1 , . FAMILY OF SCHUR MULTIPLIERS... 15
Summarizing 𝐼 ( 𝑥 ) ≤ max{ 𝐶 , 𝐶 , 𝐶 } 1 | 𝑥 | 𝛾 +1 . (cid:3) Acknowledgments.
The authors would like to thank professor A. Siskakisfor stimulating discussions on the problem. In fact his question (privatecommunication) whether
Mult( 𝓁 𝑝 ) ⊆ 𝑇 ( 𝓁 𝑝 ) or not was the initial motivationfor considering the problem. R EFERENCES [1] A. B. Aleksandrov and V.V. Peller. Hankel and Toeplitz-Schur multipliers.
Math.Ann. , 324(2):277–327, October 2002.[2] A. Aleman and A. G. Siskakis. An integral operator on 𝐻 𝑝 . Complex Var. TheoryAppl. , 28(2):149–158, October 1995.[3] A. Aleman and A. G. Siskakis. Integration operators on Bergman spaces.
IndianaUniv. Math. J. , 46(2):337–356, 1997.[4] G. Bennett. Schur multipliers.
Duke Math. J. , 44(3):603–639, 09 1977.[5] O. Blasco and I. García-Bayona. A class of Schur multipliers of matrices with operatorentries.
Mediterr. J. Math. , 16(4), June 2019.[6] A. Bonami and J. Bruna. On truncations of Hankel and Toeplitz operators.
Publ. Mat. ,43(1):235–250, 1999.[7] O. F. Brevig, K. M. Perfekt, and K. Seip. Volterra operators on Hardy spaces of Dirich-let series.
J. Reine Angew. Math. (Crelles Journal) , 2019(754):179–223, September2019.[8] G. Brown, F. Dai, and F. Mricz. The maximal Riesz, Fejér, and Cesáro operatorson real Hardy paces.
J. Fourier Anal. and Appl. , 10(1):27–50, January 2004.[9] A. P. Calderon. Commutators of singular integral operators.
Nat. Acad. Sci. ,53(5):1092–1099, May 1965.[10] R. Cheng, J. Mashreghi, and W. T. Ross. Multipliers of sequence spaces.
Concr. Oper. ,4(1):76–108, October 2017.[11] C. Coine. Schur multipliers on ( 𝑙 𝑝 , 𝑙 𝑞 ) . J. Operator Theory , 79(2):301–326, January2018.[12] K. R. Davidson and A. P. Donsig. Norms of Schur multipliers.
Illinois J. Math. ,51(3):743–766, 07 2007.[13] K. R. Davidson and V. I. Paulsen. Polynomially bounded operators.
J. Reine Angew.Math. , 487:153–170, 1997.[14] P. Galanopoulos. The Cesáro operator on Dirichlet spaces.
Acta Sci. Math. (Szeged) ,67:411âĂŞ–420, 2001.[15] M. Jevtić, D. Vukotić, and M. Arsenović.
Taylor Coefficients and Coefficient Multi-pliers of Hardy and Bergman-Type Spaces . Springer International Publishing, 2016.[16] M. Lacey and C. Thiele. 𝑙 𝑝 estimates for the bilinear Hilbert transform. Proc. Natl.Acad. Sci. USA , 94(1):33–35, January 1997.[17] J. Pau. Integration operators between Hardy spaces on the unit ball of 𝕔 𝑛 . J. Funct.Anal. , 270(1):134–176, January 2016.[18] Gilles Pisier.
Similarity Problems and Completely Bounded Maps . Springer BerlinHeidelberg, 2001.[19] Ch. Pommerenke. Schlichte funktionen und analytische funktionen von beschrÃďnk-ter mittlerer oszillation.
Comment. Math. Helv. , 52:591–602, 1977.[20] É. Ricard. On a question of Davidson and Paulsen.
J. Funct. Anal. , 192(1):283 – 294,2002. [21] F. Weisz.
Convergence and Summability of Fourier Transforms and Hardy Spaces .Springer International Publishing, 2017.D
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