Multipliers of Sequence Spaces Raymond Cheng
University of Richmond UR Scholarship Repository
Math and Computer Science Faculty Publications Math and Computer Science
2017 Multipliers of sequence spaces Raymond Cheng
Javad Mashreghi
William T. Ross University of Richmond, [email protected]
Follow this and additional works at: https://scholarship.richmond.edu/mathcs-faculty-publications Part of the Other Mathematics Commons This is a pre-publication author manuscript of the final, published article.
Recommended Citation Cheng, Raymond; Mashreghi, Javad; and Ross, William T., "Multipliers of sequence spaces" (2017). Math and Computer Science Faculty Publications. 213. https://scholarship.richmond.edu/mathcs-faculty-publications/213
This Post-print Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. Concr. Oper. 2017; 4: 76–108
Concrete Operators
Research Article
Raymond Cheng, Javad Mashreghi, and William T. Ross* Multipliers of sequence spaces https://doi.org/10.1515/conop-2017-0007 Received October 26, 2016; accepted September 5, 2017.
p Abstract: This paper is selective survey on the space `A and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality.
Keywords: Sequence spaces, Multipliers, Hardy spaces, Inner functions
MSC: 30J10, 30J15, 30H10, 30B10
1 Introduction
Sequence spaces such as the classical spaces `p play an important role in functional analysis. Indeed, they are often the first Banach spaces covered in a basic functional analysis course. Moreover, they often serve as starting points, or possibly ending points, for various conjectures. Historically, via Beurling’s theorem, the unilateral shift S on `2 was one of the first operators to have a full characterization of its invariant subspaces. They key observation by Beurling was to equivalently recast `2 from a mere sequence space to a Hilbert space of analytic functions H 2, the Hardy space, where the vast toolbox of function theory comes into play. This paper is a selective survey of results on the sequence space `p, indexed by the nonnegative integers, with p a special emphasis on the associated Banach space of analytic functions `A. Here we focus on their multipliers. 2 As it turns out, the multipliers of `A (the Hardy space) are thoroughly understood since they turn our to be just the bounded analytic functions on the open unit disk. When p 2, the multipliers are well studied but are still 6D somewhat mysterious, and some basic questions remain open. For example, every inner function is a multiplier of `2 ; however, the atomic singular inner function is not a multiplier for `p when p 2. In fact, it is unknown whether A A ¤ any singular function serves as a multiplier when p 2 (though it is known that many do not). Furthermore, since ¤ in the p 2 case the multipliers are just the bounded analytic functions, their non-tangential boundary behavior is D well understood. When 1 6 p < 2 the multipliers actually enjoy somewhat better boundary behavior than generic p q bounded analytic functions. Even more surprising is that the multipliers of `A and those of its dual space `A are the p q same set – even though the spaces `A and `A are very different in terms of their boundary behavior. Recent work is beginning to shed some light on the fact that in some Banach spaces of analytic functions, every function can be written as a quotient of two multipliers. This is indeed true for the Hardy space, and for the Dirichlet p space, as well as other reproducing kernel Hilbert spaces with a Nevanlinna-Pick kernel. For `A this “quotient of two multipliers” property turns out to be spectacularly false when p > 2, but remains an open question when p < 2. We became interested in these sequence spaces through our work in two papers [20] and [5], where we studied various natural function theory questions through the lens of Birkhoff-James orthogonality. In [5] we explored a version of the classical inner-outer factorization and its applications to ARMA processes, while in [20] we revisited some classical estimates of zeros of analytic functions. Work on those papers led us quite naturally to questions
Raymond Cheng: Old Dominion University, E-mail: [email protected] Javad Mashreghi: Université Laval, E-mail: [email protected] *Corresponding Author: William T. Ross: University of Richmond, E-mail: [email protected] Multipliers of sequence spaces 77
p about multipliers of `A; in fact we continue that discussion in this paper with a multiplier proof of a result in [20], as well as a refinement, via Birkhoff-James, of coefficient estimates of multipliers.
p 2 Basic properties of `A
For p Œ1; / define `p to be the set of sequences 2 1
a .a0; a1; : : :/ D of complex numbers for which !1=p X1 p a p ak < : k k WD j j 1 k 0 D p p The quantity a p defines a norm on ` which makes ` a Banach space. Furthermore, from Hölder’s inequality, k kp p q we know that .` / the normed dual of ` is isometrically isomorphic to ` , where q denotes the usual conjugate index, i.e., 1 1 1; (1) p C q D via the bi-linear pairing
X1 p q .a; b/ akbk; a ` ; b ` : (2) D 2 2 k 0 D Here, in the case p 1, we have q , and the dual space .`1/ ` is endowed with the norm D D 1 D 1
b sup bk k1 0: k k1 WD fj jg D Throughout this paper, we will always adhere to the notation that q is the Hölder conjugate index to p. For an a `p we set 2 X1 k a.z/ akz (3) D k 0 D to be the power series whose Taylor coefficients are a. Note the use of a (bold faced) to represent a sequence and a (not bold faced) to represent the corresponding power series. Consider the case when p .1; /. By Hölder’s inequality we see that for any z D z C z < 1 , 2 1 2 D f 2 W j j g !1=p !1=q  Ã1=q X1 k X1 p X1 kq 1 ak z 6 ak z a p : j jj j j j j j D k k 1 z q k 0 k 0 k 0 D D D j j This implies that the above power series used to define the function a in (3) determines an analytic function on D. Let us define `p a a `p A WD f W 2 g p p and endow each a `A with the norm a p. With this, `A becomes a Banach space of analytic functions on D. 2 p k k Furthermore, for each z D and a ` we have 2 2 A  1 Ã1=q a.z/ 6 a p : (4) j j k k 1 z q j j Similarly, if p 1, then D X1 k a.z/ ak z 6 a 1: j j D j jj j k k k 0 D p Thus if a sequence of functions converges in the norm of `A then it converges uniformly on compact subsets of D. p The following is obvious from the definition of `A, but worth stating here for later use. 78 R. Cheng et al.
Proposition 2.1. Let p Œ1; /. If a `p with 2 1 2 A
X1 k a.z/ akz ; D k 0 D then K X k a akz 0; K : p ! ! 1 k 0 D Corollary 2.2. If p Œ1; /, then the analytic polynomials are dense in `p. 2 1 A 1 1 p Of special distinction is the Wiener algebra `A. Let us recall that ` ` for all p Œ1; /. Furthermore, for 1  12 1 a ` , the Taylor series converges uniformly on D z C z 6 1 and thus ` is contained in C.D/, the 2 A D f 2 W j j g A continuous functions on D. We now address the “algebra” part of the term Wiener algebra. For two sequences a and b, the convolution a b is the sequence ( n ) X akbn k : k 0 n>0 D By multiplying Taylor series coefficients, notice how a b corresponds via (3) to the pointwise product a.z/b.z/ of the functions a and b. Young’s inequality [26, p. 37]
p 1 a b p 6 a p b 1; a ` ; b ` ; (5) k k k k k k 2 2 shows that `1 is a convolution algebra (i.e., a; b `1 a b `1). Now by the correspondence between 2 H) 2 1 convolution of sequences and multiplication of power series, we see that that `A is an algebra of functions (i.e., a; b `1 ab `1 ). 2 A H) 2 A
3 Evaluation functionals and duality
The estimate in (4) says that for each w D, the evaluation functional 2 p ƒw ` C W A ! is continuous for each p .1; /. 2 1 We can even compute its norm
p ƒw sup f .w/ f ` ; f p 6 1 : k k D fj j W 2 A k k g Proposition 3.1. Let p Œ1; /. For each w D, 2 1 2  1 Ã1=q ƒw : k k D 1 w q j j Proof. From (4) we get 1 ƒw 6 : (6) k k .1 w q/1=q j j For fixed p .1; /, consider the test function 2 1 1 X1 f .z/ . w q 2w/nzn D 1 w q 2wz D j j n 0 j j D and observe that 1 ƒwf f .w/ : D D 1 w q j j Multipliers of sequence spaces 79
On the other hand,
X1 X1 X1 1 f p w q 2w pn w .q 1/pn w qn : k kp D jj j j D j j D j j D 1 w q n 0 n 0 n 0 D D D j j Hence ƒ f 1 > w ƒw j j q 1=q : (7) k k f p D .1 w / k k j j Comparing (6) and (7), we deduce that 1 ƒw : (8) k k D .1 w q/1=q j j The p 1 case is similarly handled. D p p p With our identification of `A with ` , we can appeal to (2) and see that the norm dual of `A can be isometrically q identified with `A via the bi-linear pairing
X1 p q .a; b/ akbk; a ` ; b ` : D 2 A 2 A k 0 D from (2). Since this series converges absolutely, we know from either Abel’s Theorem or the Dominated Convergence Theorem that X1 2k .a; b/ lim akbkr : D r 1 ! k 0 D Now an integral calculation, and the simple fact that
2 Z ik d e ık;0; 2 D 0 shows that we can write the pairing in terms of the “Cauchy pairing”
2 Z i i d .a; b/ lim a.re /b.re / : D r 1 2 ! 0
Using the notion of duality, the following is another useful interpretation of the evaluation functional ƒw, which also yields a more concise proof of Proposition 3.1. For a fixed w D, define 2
X1 n n kw.z/ w z : (9) WD n 0 D q Clearly kw ` and, by (2), 2 A ƒwf .f; kw/: (10) D In other words, kw plays the role of a reproducing kernel.
4 Connection to Hardy spaces
p For p Œ1; / the classical Hardy space H is the space of analytic functions f on D for which 2 1 1=p 0 2 1 Z i p d f H p @ sup f .re / A < : (11) k k D 0 For p , the class H 1 is the space of bounded analytic functions on D. Classical theory [7] says that functions D 1 in H p have radial limits f .ei / lim f .rei / WD r 1 ! for almost every Œ0; 2 and the corresponding (almost everywhere defined) boundary function ei f .ei / p 2 7! belongs to L .T; d/. In fact, 1=p 0 2 1 Z i p d f H p @ f .e / A : k k D j j 2 0 When p 2, the space H 2 can also be described as those analytic functions D X1 k f .z/ akz D k 0 D on D for which X1 2 ak < : j j 1 k 0 D Moreover, by Parseval’s identity, we have 2 Z 2 i 2 d X1 2 f 2 f .e / ak < : k kH D j j 2 D j j 1 k 0 0 D Consequently, `2 H 2 with equal norms, and thus H 2 is a Hilbert space. A D The Hausdorff-Young inequalities [26, p. 101] show, for p Œ1; 2, that H p `q with 2  A p a q 6 a H p a H : k k k k 2 On the other hand, when p Œ1; 2, we also have `p H q with 2 A  p a H q 6 a p a ` : k k k k 2 A In particular, when p Œ1; 2, every a `p has radial boundary values almost everywhere. When p .2; / the 2 2 A 2 1 above containments fail, and, as we will see in the next section, radial limits become a problem as well. 5 Boundary values p q q From the discussion in the previous discussion, `A H , when p Œ1; 2. Furthermore, each function in H , and p  2 hence `A, has a radial limit almost everywhere on T. When p > 2, the boundary behavior can be more complicated. To see this, we bring in a theorem of Littlewood. Proposition 5.1 (Littlewood [7]). Assume that an n>0 is a sequence of complex numbers such that f g X1 2 an : j j D 1 n 0 D Set X1 n f .z/ "nanz ; D n 0 D where "n 1; 1 . Then there are choices of signs "n such that the corresponding function f fails to have radial 2 f g limits almost everywhere on T. Multipliers of sequence spaces 81 As a matter of fact, Littlewood introduces a probability measure on the family of all possible signs, i.e., 1; 1 0 , f g@ and then he shows that for almost all such signs the above proposition holds. However, for our discussion below, the existence of even one such function is enough. Let N denote the Nevanlinna class of analytic functions on D which can be written as the quotient of two bounded analytic functions. By well-known theorems of Riesz and Fatou [7], every function in the Nevanlinna class has a radial limit almost everywhere. Corollary 5.2. For each p .2; / the space `p is not contained in N . 2 1 A p Proof. Pick any sequence an n>0 ` such that f g 2 X1 2 an : j j D 1 n 0 D For example, 1 > an 2 ; n 1; D p 2 n C does the job. Then, by Proposition 5.1, there are choices of signs "n such that the corresponding function f fails to p have radial limits almost everywhere on T. This function certainly belongs to `A but not the Nevanlinna class N . p 6 Operators on `A We define the forward shift operator p p S ` ` ;Sa .0; a0; a1; a2; : : :/ W ! D and observe that S is an isometry on `p. For a `p, let Œa be the S-invariant subspace generated by a, that is, 2 _ Œa a;Sa;S 2a;::: ; WD f g where W denotes the closed linear span in `p. A vector a `p is said to be cyclic if Œa `p. 2 D Also define the backward shift operator q q S ` ` ;S a .a1; a2; : : :/ W ! D and observe that S is a contraction on `q. If a `p and b `q, it is straightforward to see that 2 2 .Sa; b/ .a;S b/: (12) D This is a fundamental connection between S and S . p One can view the shift S on `A as the operator a.z/ za.z/ 7! p of multiplication by the independent variable z on the corresponding function space `A. From this viewpoint, note p p that for a `A, Œa is the `A-closure of the set of all P a, where P is an analytic polynomial. We will identify the 2 p p shift operator on the sequence space ` with the multiplication (by z) operator on the function space `A, and denote both by S. A similar convention is applied for S . For example, considering the definition (9), we see that S kw wkw: (13) D Again, the reader should be take note of the absence of conjugation in the above formula. The S-invariant subspaces of `p, i.e., those (closed) subspaces M `p for which SM M , are sometimes A  A  difficult to describe. When p 2, we have already seen that `2 H 2 and a well-known theorem of Beurling D A D 82 R. Cheng et al. [7] says that the (non-trivial) invariant subspaces M of `2 H 2 are given by M ‚H 2, where ‚ is an inner A D D function. Notice how this says that the quotient space M=SM is one dimensional. When p > 2, the situation is much more complicated. For example, one can show [1] that given any n N there is an S-invariant subspace M p 2 [ f1g of `A such that the quotient space M=SM is n-dimensional. Given w D, the difference quotient operator Qw is defined on the set of analytic functions on D by 2 f .z/ f .w/ .Qwf /.z/ : WD z w Proposition 6.1. Let w D. Then 2 p p f ` Qwf ` : 2 A H) 2 A p Moreover, Qw is a bounded operator on `A. In particular, Q0 is precisely the backward shift S . For the proof of the p 2 case see [11, p. 100]. The general D case is substantially similar. p In other words, one can always “divide out” a zero of a and still remain in `A. For many Banach spaces of analytic functions contained in the Nevanlinna class, the most prominent example being the H p classes, one can divide out any inner factor and still remain in the space. Moreover if ‚ is inner then ‚f H p f H p: 2 H) 2 Even though `p is contained in H q for p .1; 2/, `p does not always have the analogous property. We will discuss A 2 A this further in the next section about multipliers. 7 Multipliers p An analytic function ' on D is called a multiplier of `A if f `p 'f `p: 2 A H) 2 A p p p0 The set of multipliers of `A will be denoted by Mp. (One can also consider a multiplier ' from `A to `A , i.e., p p f ` 'f ` 0 ; 2 A H) 2 A though we will not discuss this in our current paper.) For ' Mp, an application of the closed graph theorem shows that the operator 2 p p M' ` ` ;M' f 'f W A ! A D is continuous. Thus we define the multiplier norm of ' by p ' sup 'f p f ` ; f p 6 1 : k kMp WD fk k W 2 A k k g p In other words, the multiplier norm of ' coincides with the operator norm of M' on `A. It is often customary to equate the multiplier ' with the multiplication operator M' . Proposition 7.1. Let p .1; /. If ' Mp then ' is a bounded function and 2 1 2 6 sup '.z/ z D ' Mp : fj j W 2 g k k n Proof. Since ' Mp then so is ' for all n N and thus, for each z D, we can use (6) to see that 2 2 2 n n ' n n n ' M' .1/ Mp '.z/ ƒz' 6 k k k k k k : j j D j j .1 z q/1=q D .1 z q/1=q D .1 z q/1=q j j j j j j Taking the n-th root and letting n yields the result. ! 1 Multipliers of sequence spaces 83 p The above result, and the fact that the constant functions belong to `A, imply that p Mp H ` ; (14)  1 \ A 1 whenever p .1; /. When p 1, Young’s inequality (5) ensures that `A, the Wiener algebra, coincides with the 2 1 1 D algebra of multipliers on `A. In other words, 1 M1 ` : D A 2 2 When p 2, we use the fact that `A is the Hardy space H and the boundedness of integral means in the definition 2 D of H from (11) to show that H M2. Hence 1  M2 H : D 1 We will see from Corollary 15.7 below that the singular inner function 1 z s.z/ exp. C /; D 1 z which is certainly bounded on D, is not a member of Mp for any p 1; 2. Thus 6D p Mp ¨ ` H ; p 1; 2: A \ 1 6D 8 Mp as the commutanat p Clearly '.z/ z is a multiplier of ` . In fact Mz S and we have already established that S is an isometric D A D operator. Moreover, since M' S SM' for all ' Mp, we see that Mp is a subset of the commutant of S, defined D 2 by p S A B.` / AS SA : f g0 D f 2 A W D g In fact, more can be said. Proposition 8.1 (Nikolskii [21]). For p Œ1; / we have S Mp. 2 1 f g0 D Proof. Clearly we have Mp S . Conversely suppose A S . Then for any analytic polynomial P we have  f g0 2 f g0 A.P.S1// P.S/A.1/; D p p equivalently, A.P / PA.1/. By the density of the polynomials in `A (Proposition 2.1) we can, for a given f `A, D p 2 find a sequence of polynomials Pn n>1 such that Pn f in the norm of `A. Since point evaluations on D are p f g ! continuous in the norm of ` (Proposition 3.1), we get Pn f pointwise on D. Since APn Af both in norm A ! ! as well as pointwise on D, it follows that Af A.1/f . Thus ' A.1/ Mp and A M' . D D 2 D To bring in more operator theory techniques, we now give an equivalent characterization of Mp explored in [21]. Given a sequence of complex numbers an n>0, define the infinite (Toeplitz) matrix A by f g 0 1 a0 0 0 0 Ba a 0 0 C B 1 0 C B C A Ba2 a1 a0 0 C : (15) WD B C a3 a2 a1 a0 B C @ : : : : : A : : : : :: Proposition 8.2 (Nikolskii [21]). Suppose X1 n '.z/ anz D n 0 D is an analytic function on D. Then ' Mp if and only if the infinite matrix A from (15) defines a bounded operator 2 on `p. In this case, p p ' Mp A ` ` : k k D k k ! 84 R. Cheng et al. Proof. Let X1 n p f .z/ bnz ` D 2 A n 0 D p and write b bn n>0 ` . Then WD f g 2 n X1 n X .'f /.z/ cnz ; cn akbn k: D D n 0 k 0 D D Therefore ' Mp if and only if 2 ˇ n ˇp X1 p X1 ˇ X ˇ cn ˇ akbn kˇ < : j j D ˇ ˇ 1 n 0 n 0 ˇk 0 ˇ D D D On the other hand, setting c cn n>0, we see that b and c are related via the matrix identity WD f g Ab c: (16) D p Thus ' is a multiplier for `A if and only if b `p Ab `p: 2 ” 2 By the closed graph theorem, the latter is equivalent to A B.`p/. 2 For the equality of norms, note that by (16), p ' sup 'f p f ` ; f p 6 1 k kMp D fk k W 2 A k k g sup Ab p b p 6 1 D fk k W k k g A `p `p : D k k ! From (2) recall the bi-linear pairing .a; b/ between `p and `q. Proposition 8.2 implies the useful inequality p q .Ax; y/ 6 ' x p y q; x ` ; y ` : (17) j j k kMp k k k k 2 2 This next result relates the multipliers on the dual spaces and is often an important reduction to our multiplier discussion. Proposition 8.3 (Nikolskii [21]). For p .1; / we have Mp Mq with equal multiplier norms. 2 1 D p p Proof. Extend the definition of ` ` .N0/ to D 8 !1=p 9 p < X p = ` .Z/ b bn n Z b p bn < : WD D f g 2 W k k D j j 1 : n Z ; 2 p p Extend the definition of the shift S on ` to ` .Z/ as S bn n Z bn 1 n Z f g 2 D f g 2 p and the backward shift B on ` .Z/ as B bn n Z bn 1 n Z: f g 2 D f C g 2 p p For p .1; / the projection P ` .Z/ ` .N0/ defined by 2 1 W ! P bn n Z bn n N0 f g 2 D f g 2 is continuous (in fact contractive). If ' Mp with 2 X1 n '.z/ anz D n 0 D Multipliers of sequence spaces 85 p p and A ` .N0/ ` .N0/ from (15) is the matrix operator formed from the sequence an n N0 , one can use the W ! p f g 2 facts that the operators S and B are isometric on ` .Z/ and apply the Banach-Steinhaus Theorem to see that the sequence of operators N N B APS ;N N0; 2 are uniformly bounded in operator norm. Applying this sequence of operators to the basis vectors en ıj;n j Z D f g 2 one can verify (equating the sequence an n N0 with the sequence an n Z, where an 0 for n < 0) that for any p f g 2 f g 2 D bn n Z ` .Z/, f g 2 2 ( ) N N X lim .B APS / bn n Z bkan k N f g 2 D !1 k Z n 2 Z p 2 and this operator, which we call L, is continuous on ` .Z/. Informally, L is multiplication by ' when equating a p sequence in ` .Z/ with its corresponding Fourier series. By the dual pairing X1 p q .c; d/ ckdk; c ` .Z/; d ` .Z/; D 2 2 k Z 2 q one can show that the adjoint L of L, which is continuous on ` .Z/, turns out to be ( ) X L bn n Z bkan k ; f g 2 D e k Z n Z 2 2 where an n Z : : : ; a3; a2; a1; a0; 0; 0; 0; : : : : fe g 2 D f g i q Informally, one can think of L is multiplication by '.e / on the Fourier series formed by sequences from ` .Z/. q q Restricting L to ` . N0/ one can see, by reindexing, that ' is a multiplier on ` . Thus we have shown that A Mp Mq. Since this argument was symmetric in p and q, we conclude that Mp Mq.  D The proof also shows that M' `p `p M' `q `q : k k A ! A D k k A! A Corollary 8.4. If X1 k ' akz Mp D 2 k 0 D then p q (i) ' `A `A, 2 \ 6 (ii) max ' p; ' q ' Mp , fk k k k g k k 1 (iii) a0 a1 a2 an 6 ' .n 1/ q . j j C j j C j j C C j j k kMp C Proof. Statement .i/ follows from Proposition 8.3. Statement .ii/ follows from ' p ' 1 p Mp1 p 6 ' 1 p ' k k D k k D k k k kMp k k D k kMp and Proposition 8.3. To prove .iii/, apply i arg aj x .1; 0; 0; : : : /; y .0; : : : ; n; 0; 0; : : : /; j e D D D to (17) to deduce a0 a1 a2 an 0a0 nan j j C j j C j j C C j j D j C C j .Ax; y/ D j j 1 6 ' .n 1/ q : k kMp C We remark that when examining Mp, we can use Proposition 8.3 to justify focusing our efforts to the study of Mp for p .1; 2. 2 86 R. Cheng et al. 9 Connection to Fourier multipliers 2 For p Œ1; 2 let Ap.T/ denote the space of all functions f in the Lebesgue space L .T/ whose Fourier coefficients 2 2 Z i in d fb .n/ f .e /e ; n Z; WD 2 2 0 p form a sequence in ` .Z/. More explicitly, ( ) 2 X1 p Ap.T/ f L .T/ fb .n/ < : (18) WD 2 W n j j 1 D 1 p Parallel to the definition of the norm in `A, we define a norm 1 ! p X1 p f Ap .T/ fb .n/ k k WD n j j D 1 p 2 on Ap.T/. Use that fact ` ` for p Œ1; 2 along with Parseval’s Theorem to see that Ap.T/ is a Banach space  2 when p Œ1; 2. 2 The corresponding multiplier space, the so called `p-Fourier multipliers, is defined to be the family of all ' 2 L1.T/ (essentially bounded Lebesgue measurable functions on T) such that 'f Ap.T/ whenever f Ap.T/, 2 2 i.e., Mp.T/ ' L1.T/ f Ap.T/ 'f Ap.T/ : WD f 2 W 2 W 2 g We naturally define the corresponding multiplier norm by ' M . / sup 'f A . / f A . / 6 1 : k k p T WD fk k p T W k k p T g In Proposition 7.1, we saw that Mp H . Therefore each ' Mp can be considered either as an analytic  1 2 function on D or, via radial boundary values, as a measurable function on T with a vanishing negative spectrum (i.e., its Fourier coefficients with negative indices are zero). To distinguish between the two interpretations, we denote the latter function, via radial boundary values, by '. It turns out that Mp naturally sits inside Mp.T/. Proposition 9.1. Let p Œ1; 2 and ' H . Then 2 2 1 ' Mp ' Mp.T/: 2 ” 2 Moreover, ' ' M . /. k kMp D k k p T p 2 2 2 p Proof. Let ' Mp.T/. Since `A `A H L .T/, we can consider `A as a subclass of Ap.T/. In particular, p2  D  for each f ` we have 'f Ap.T/ and 2 A 2 ' f A . / 6 ' M . / f A . /: k k p T k k p T k k p T Moreover, the negative parts of the spectrum of ' and f are identically zero, and thus so is that of 'f . p Therefore, 'f Ap.T/ implies that 'f ` and the above inequality can be rewritten as 2 2 A 'f p 6 ' M . / f p: k k k k p T k k 6 In other words, ' Mp and ' Mp ' Mp .T/. 2 k k k k iN i Now assume that ' Mp. Let f be a trigonometric polynomial of degree N . Then e f .e / is an analytic 2 polynomial and can be considered as an element of `p. Thus, since ' is a multiplier of `p, we have 'eiN f `p A A 2 A and iN iN 'e f p 6 ' e f p: k k k kMp k k Multipliers of sequence spaces 87 iN Since the operator f e f is isometric on Ap.T/, we can rewrite the previous inequality as 7! ' f A . / 6 ' f A . /: k k p T k kMp k k p T Since trigonometric polynomials are dense in Ap.T/, the estimate above holds for all f Ap.T/. Therefore ' 6 2 2 Mp.T/ and ' Mp . / ' Mp . Combine this inequality with the reverse of it shown before to complete the k k T k k proof. Though this Fourier multiplier problem might seem like a detour from our main discussion concerning the multipliers p of `A, it will surface again later when examining boundary values of multipliers. 10 Quotients of multipliers 2 Classical factorization results [7] say that any f H can be written as f h1=h2 where h1 and h2 are bounded 2 2 D analytic functions and h2 is zero free. Since the multiplier space of H is precisely H 1, this result can be stated in the following equivalent form for a Banach space of analytic functions X and its multiplier space M, i.e., h1 f ; (19) D h2 where h1 and h2 belong to the multiplier space M and h2 is zero free. This point of view opens the door for the same question about any Banach space of analytic functions. In some cases the answer is known. Besides the Hardy space H 2, it seems that the answer is affirmative for the classical Dirichlet space as well as reproducing kernel Hilbert p spaces with a Nevanlinna-Pick kernel (several unpublished results). We are interested in this question for `A spaces. When p 2, we are in the classical setting of `2 H 2 and thus, as seen above, the answer is affirmative. The case D A D p 1 is also trivial since `1 is itself an algebra, and thus it coincides with its multiplier algebra. For p .1; 2/, the D A 2 question is still open. Using function theory tools mentioned earlier, we can show that the answer is negative when p .2; /. 2 1 Corollary 10.1. Let p .2; /. Then there are functions in `p which cannot be represented as the quotient of two 2 1 A multipliers. p Proof. By Corollary 5.2, there are functions in `A which are not in the Nevanlinna class N . Such a function cannot be represented as the quotient of two multipliers, since by (14), such a quotient is in N . Using a similar technique, one can show that the representation (19) fails in the Bergman space since it is well-known that the Bergman space is not contained in the Nevanlinna class. 11 Isometric multipliers Which multipliers ' satisfy p 'f p f p; f ` ‹ (20) k k D k k 2 A These are known as the isometric multipliers. Once again the case p 2 is exceptional. It is well known that (20) holds for `2 H 2 if and only if ' is an D A D inner function. Indeed, Proposition 7.1 and (20) show that '.z/ 6 1; z D: j j 2 Then use Parseval’s identity to rewrite (20) in integral form as 2 Z d f .ei / 2 .1 '.ei / 2/ 0; f H 2; j j j j 2 D 2 0 88 R. Cheng et al. which holds if and only if '.ei / 1 almost everywhere. In other words, ' is inner. When p 2 the story is j j D 6D different. Observe that n p z f p f p; f ` ; (21) k k D k k 2 A > n p for all n 0. In other words, the monomials z are isometric multipliers for `A. Are there others? Theorem 11.1 (Nikolskii [21]). If p .1; / 2 and ' is an isometric multiplier for `p, then '.z/ zn for 2 1 n f g A D some n > 0 and a unimodular constant . The proof is based on two sets of elementary inequalities due to Bernoulli [22, p. 31]. In the following x > 0, y > 0 and t > 1. (i) For 0 < ˛ < 1 (a) we have .x y/˛ 6 x˛ y˛ C C and equality holds if and only if either x 0 or y 0. D D (b) we have .1 t/˛ 6 1 ˛t C C and equality holds if and only if t 0. D (ii) For 1 < ˛ < 1 (a) we have .x y/˛ > x˛ y˛ C C and equality holds if and only if either x 0 or y 0. D D (b) we have .1 t/˛ > 1 ˛t C C and equality holds if and only if t 0. D Proof of Theorem 11.1. We treat the case 1 < p < 2 for which the first set of inequalities above are used. The other case is similar. n We may write '.z/ z '1.z/, where is unimodular, n is the order of the zero of h at the origin, and '1 is D such that '1.0/ > 0. By (21), we have p '1f p f p; f ` : k k D k k 2 A We will now show that '1 1. Á p Hence, considering the above reduction, assume that '.0/ > 0 and 'f p f p for all f ` . Take k k D k k 2 A f .z/ 1 ei z; D C where we treat as a free parameter. If X1 n '.z/ anz ; D n 0 D the isometric identity can be rewritten as p X1 i p a0 an 1 ane 2: (22) j j C j C C j D n 0 D We integrate both sides with respect to d. First, due to periodicity, we have 2 2 Z Z i p ˇ i ˇp a be d ˇ a b e ˇ d: j C j D j j C j j 0 0 Multipliers of sequence spaces 89 Second, an elementary calculation reveals that ˇ i ˇ2 2 2 2 2 ˇ a b e ˇ a b 2 ab cos . a b / .1 s cos /; j j C j j D j j C j j C j j D j j C j j C where 2 ab s j j : D a 2 b 2 j j C j j Note that 0 6 s 6 1 and thus t s cos > 1. Third, by the above Bernoulli inequalities we have WD 2 2 Z 2 2 p Z 1 ˇ i ˇp . a b / 2 p ˇ a b e ˇ d j j C j j .1 s cos / 2 d 2 j j C j j D 2 C 0 0 2 a p b p Z ps Á 6 j j C j j 1 cos d 2 C 2 0 a p b p; D j j C j j and the equality holds if and only if either a 0 or b 0. Returning to (22), we get D D p X1 p p X1 p 2 6 a0 . an 1 an / 2 an : j j C j C j C j j D j j n 0 n 0 D D If we plug f 1 in we see that D !1=p X1 p ' p an 1: k k D j j D n 0 D Hence, in the above relation, equality holds, which in return implies that equality holds in all preceding inequalities. Since a0 0, we must have ¤ a1 0: D We now repeat the above procedure with the function f .z/ 1 ei z2 D C P p and deduce that a2 0. By induction, we have an 0 for all n > 1. Since a0 > 0 and n1 0 an 1, we D D D j j D conclude that ' a0 1. Á D 12 Smooth multipliers The family of analytic functions which are defined on a disk larger than the open unit disc is denoted by Hol.D/. From Young’s Inequality (see (5)) we see that 1 ` Mp; p Œ1; / A  2 1 with equality when p 1. Thus certainly we have D Hol.D/ Mp:  We present below an alternative proof of this fact where we can obtain further information. Let us recall Schur’s test. Let A Œaij be an infinite matrix, and let p .1; /. Assume that there are positive D 2 1 constants ˛ and ˇ and positive sequences pi and qj such that f g f g X p p aij p 6 ˛ q ; j > 1; j j i j i 90 R. Cheng et al. and X q q aij q 6 ˇ p ; i > 1: j j j i j Then A is a bounded operator on `p and moreover, 1 1 A `p `p 6 ˛ p ˇ q : (23) k k ! For each f Hol.D/ with Taylor series expansion 2 X1 n f .z/ anz ; D n 0 D there is R > 1 and a c > 0 such that c an 6 ; n > 0: (24) j j Rn The constants R and c depend on f , but work uniformly with respect to n. This exponential decay plays the major role in establishing the following result. Theorem 12.1. If p .1; /, then Hol.D/ Mp. 2 1  Proof. By Proposition 8.2, it is enough to show that the matrix A formed with the coefficients of ', according to p i recipe (15), is a bounded operator on ` . We apply Schur’s test with pi qi t , where t is a positive parameter D D to be determined momentarily. Fixing j , by (24), we have X p X1 ip aij pi ai j t j j D j j i i j D ip X1 t 6 c Ri j i j D  p Ãi X1 t ct jp D R i 0 D c p p q ; j > 0: D 1 t j R Similarly, fixing i, by (24), we have i X q X jq aij qj ai j t j j D j j j j 0 D i X t jq 6 c Ri j j 0 D i j X  1 à ct iq D t qR j 0 c D pq; i > 0: D 1 1 i tq R Schur’s test ensures that 1 1 ! p ! q c c p p 6 A ` ` tp 1 : k k ! 1 1 R tq R The above geometric series are convergent provided that t p < R and t q > 1=R. Hence, the acceptable range of t is 1 < t < R1=p: R1=q Multipliers of sequence spaces 91 Therefore we can say that 1 1 ! p ! q c c A `p `p 6 inf p : (25) 1=q 1=p t 1 k k ! t .R ;R / 1 1 q 2 R t R In particular, with t 1 we get D c p p 6 A ` ` 1 ; (26) k k ! 1 R which is enough for our applications. In the above proof, we took t 1 in (25). Is it possible to get a better bound by choosing another value of t? In other D words, what is the optimal value of t? The proof of Theorem 12.1 contains more information than presented in the theorem. By a closer look, we obtain the following interesting convergence result. Corollary 12.2. For p .1; / and ' Hol.D/ and denote its Taylor polynomial of degree n by 'n. Then 2 1 2 lim 'n ' 0: n k kMp D !1 Moreover, the rate of decay is exponential. Proof. Since X1 k n 1 X1 k '.z/ 'n.z/ akz z C ak n 1z ; D D C C k n 1 k 0 D C D by (21), we have X1 ' ' a zk : n Mp k n 1 k k D C C k 0 Mp D Thus, according to Proposition 8.2, p p 'n ' Mp An ` ` ; k k D k k ! where the matrix An is given by (15) but with the sequence ˛k ak n 1; k > 0: D C C By (24) we have the estimate c .c=Rn 1/ 6 C > ˛k ak n 1 k n 1 k ; k 0: j j D j C C j R C C D R Therefore by (26), n 1 c=R C p p 6 An ` ` 1 ; k k ! 1 R which reveals that 'n ' exponentially decreases to zero. k kMp Corollary 12.2 does not hold for an arbitrary multiplier. For example, we saw that M2 H . However if ' M2 D 1 2 satisfies 'n ' M2 'n ' 0; k k D k k1 ! then ' is continuous on D. 92 R. Cheng et al. 1 13 `A embeds contractively in Mp In this Section, we provide a result which contains Theorem 12.1 as a special case. However, we stated that theorem separately since the estimate in (25) should provide a better bound for the norm of the narrower class of multipliers Hol.D/. 1 Theorem 13.1. For p .1; / we have ` Mp and 2 1 A  h 6 h 1 k kMp k k for every h `1 . 2 A Proof. Of course the result follows immediately from (5) (and also observed in [21]) but we include a proof using the tools developed above. We again appeal to Proposition 8.2. Hence, it is enough to show that the matrix A formed with the coefficients of h according to recipe (15) is a contraction on `p. We apply the simplest version of Schur’s test, i.e., with pi qi 1. D D Fixing j , we have X p X1 X1 aij pi ai j ak h 1: j j D j j D j j D k k i i j k 0 D D Similarly, fixing i, we obtain i i X q X X 6 aij qj ai j ak h 1: j j D j j D j j k k j j 0 k 0 D D Therefore we may take ˛ ˇ h 1: D D k k Schur’s test (23) ensures that A `p `p 6 h 1: k k ! k k 1 Corollary 13.2. Let p .1; / and for h ` denote its Taylor polynomial of degree n by hn. Then 2 1 2 A lim hn h 0: n k kMp D !1 If the coefficients of h are all nonnegative, then Theorem 13.1 is reversible. 1 Theorem 13.3. Let p .1; /. If h Mp and the Taylor coefficients of h are nonnegative, then h ` . 2 1 2 2 A Proof. In the inequality (17), take x y .1; 1; : : : ; 1; 0; 0; : : : / D D to get n i X X aj 6 .n 1/ h : C k kMp i 0j 0 D D After rearranging the sums we obtain n X  k à 1 ak 6 h : n 1 k kMp k 0 D C By the monotone convergence theorem, let n to deduce ! 1 X1 ak 6 h : k kMp k 0 D Multipliers of sequence spaces 93 14 Boundary properties of multipliers Since Mp H each h Mp has a non-tangential limit  1 2 ^ lim h.z/ z ei ! for almost every . Lebedev and Olevskii [14, 17, 18] observed that more can be said. Their work is based on the following technical discussion. Recall the definition of Ap.T/ for p Œ1; 2 from (18) 2 ( ) 2 X1 p Ap.T/ f L .T/ fb .n/ < WD 2 W n j j 1 D 1 and its corresponding space of multipliers Mp.T/ ' L1.T/ f Ap.T/ 'f Ap.T/ : WD f 2 W 2 W 2 g This next result is a bit technical and we refer the reader to the references for the proof. Theorem 14.1 (Lebedev and Olevskii [14, 17, 18]). If p Œ1; 2/ and Mp.T/, there is a continuous function ‰ 2 2 on T such that ‰ almost everywhere on T. D Corollary 14.2. Let p Œ1; 2/ and ' Mp. Then the unrestricted limit 2 2 lim '.z/ z ei ! exists for almost all . Proof. From Proposition 9.1 we know that the almost everywhere defined radial boundary function ' belongs to Mp.T/ and, by Theorem 14.1, there is a continuous function ˆ on T that is equal to ' almost everywhere. Since ' is the radial boundary function for ', we have the well-known Poisson integral formula 2 2 Z Z i i d i i d '.z/ Pz.e /' .e / Pz.e /ˆ.e / ; D 2 D 2 0 0 i where Pz.e / is the Poisson kernel. If eit is a point of continuity of ‰, a well-known fact from harmonic analysis says that 2 Z i i d it lim Pz.e /ˆ.e / ˆ.e /: z eit 2 D ! 0 However, almost every point of T is a point of continuity of ˆ and the result now follows. The essential feature of Corollary 14.2 is that z D freely tends to the boundary point and it is not obliged to stay 2 in a Stolz domain (non-tangential approach region). We also mention that it is sometimes (often?) the case the set of multipliers for a Banach space of analytic functions are better behaved near the boundary than generic functions in the space. 15 Inner multipliers A worthy set of functions to test as possible multipliers for Mp are the inner functions. We know that the monomials n 'n.z/ z are multipliers, in fact isometric multipliers. Are there any other inner multipliers? D Certainly any finite Blaschke product is an inner multiplier since these are analytic in an open neighborhood of p D (see Theorem 12.1). There are some infinite Blaschke products which are multipliers for all of the `A classes. 94 R. Cheng et al. Theorem 15.1 (Vinogradov [25], Verbitskii [23]). Let zn n>1 be a Blaschke sequence in D such that f g (i) limn zn 1, !1 D (ii) and X 1 zn O."/; " 0: j j D ! n 1 zn <" f Wj j g Let B be the corresponding Blaschke product. Then \ B Mp: 2 1 .r; ˛/ z z < r z arg.z 1/ 6 ˛ ; D f W j j g n f W j j g where r > 1 and 0 6 ˛ < =2. Theorem 15.2 (Vinogradov [24]). For each r > 1 and ˛ .0; /, we have 2 2 \ H .r; ˛/ Mp: 1  p .1; / 2 1 A little thought will show that any Blaschke product whose zeros tend non-tangentially to 1 (i.e., lie in a fixed Stolz domain) satisfies the condition of Theorem 15.2 for some r > 1 and ˛ Œ0; ; 2/, and thus the weak version 2 of Theorem 15.1 follows. However, there is a large family of functions which are not inner and still fulfill the requirements of Theorem 15.2. Note that this result does not apply to the singular inner functions  1 z à sa.z/ exp a C ; .a > 0/: WD 1 z As a matter of fact, we will see that sa does not belong to Mp, for any p .1; / 2 . 2 1 n f g We can use Corollary 14.2 to eliminate certain classes of inner functions as multipliers. Let us recall that if h is an inner function, then its boundary spectrum is n o .h/ T lim inf h.z/ 0 : WD 2 W z j j D ! Equivalently [11, p. 154], if h BS is the decomposition of h as the product of a Blaschke factor B formed with D zeros zn n>1 and a singular inner function S formed with the singular measure , then .h/ is precisely the union f g of the spectrum of and the accumulation points of zn n>1 on T. The Lebesgue measure of a measurable set f g E T is denoted by E .  j j Theorem 15.3. Let h be an inner function such that .h/ > 0. Then j j h Mp 62 for any p .1; / 2 . 2 1 n f g Multipliers of sequence spaces 95 Proof. Since Mp Mq (Proposition 8.3) we can assume p Œ1; 2/. If h Mp, then, by Corollary 14.2, D 2 2 h./ 0 D for almost all .h/. Since at the same time h 1 almost everywhere on T, we conclude that the Lebesgue 2 j j D measure of .h/ is zero. Surprisingly, the mere existence of a singular inner function in Mp is still an open question. According to Theorem 15.3, the condition .h/ 0 is necessary for being a multiplier. However, this condition is not sufficient. For j j D example, as we mentioned before, Verbitskii [23] showed that the simplest singular inner function  1 z à sa.z/ exp a C ; a > 0: D 1 z is not in any Mp, for p .1; / 2 . We provide some further results clarifying the fact that the condition 2 1 n f g .h/ 0 is far from being sufficient for multipliers. j j D Let E T. For an arc I T, we define the quantity   dE .I / sup J J is an arc, J I;J E : WD fj j W  \ D ;g In other words, and naively speaking, when we remove the points of E from I, then dE .I / is the size of largest arc among the remaining pieces. We say that T is a point of thickness of E provided that dE .I / o. I / for arcs I 2 D j j containing and shrinking to this point. The set of all points of thickness of E is denoted by Eth. It is clear that for a closed set E, we have Eth E. The following result establishes the connection between the measures of an arc  and dE .I /. Lemma 15.4. Let S be a singular inner function, corresponding to the singular measure , let E be its support. Suppose that S Mp for some p with p .1; / 2 . Then for each arc I T with dE .I / > 0 we have 2 2 1 n f g  I 4 6 .I / c j3 j ; dE .I / where c does not depend on I . We need a technical result first which uses our previous Fourier multiplier discussion. Proposition 15.5. Let ‚.t/ be a real-valued 2-periodic function on R such that i‚ ' e Mp.T/ WD 2 for some p .1; / 2 . Suppose I R is an interval of length at most 2 and that ‚ has a continuous derivative 2 1 nf g  of order n > 2 on I . Then c inf ‚.n/.t/ 6 ; t I j j I n 2 j j where the constant c c.‚; p; n/ does not depend on I . D Proof. By Corollary 8.3, we have Mp Mq. Therefore, without loss of generality, we assume that p < 2. Let I D denote the characteristic function of I . It is easy to directly verify that I Ap.T/ and estimate its norm. In fact, 2 the Fourier coefficient of I are given by I I .0/ j j c D 2 and sin.k I =2/ I .k/ k j j ; k Z 0 ; c D k 2 n f g where k is a unimodular constant. Hence, I Ap.T/ and, moreover, 2 I A . / I `p . / k k p T D kck Z 96 R. Cheng et al. 1 ! p X1 p I .k/ D jc j k D 1 1 0 1 p X p > @ I .k/ A jc j k 6=2 I j j j j 1 0 1 p X p > @ . I =4/ A j j k 6=2 I j j j j 1  à p I p 1 1 1 > . j j / > I p : I 4 15j j j j Note that the last estimate also holds for p . Since ' Mp.T/, and thus D 1 2 I A . / ' ' I A . / 6 ' M . / ' I A . /; k k p T D k N k p T k k p T k N k p T we conclude that 1 1 1 > p ' I Ap .T/ I : (27) k N k 15 ' M . / j j k k p T Therefore, we need a strategy to deal with the left side of above inequality. To this end, we need the well-known Corput lemma [3, Chapter I]: if a real-valued function g has a continuous derivative of order n > 2 on an interval I , then ˇ ˇ ˇZ ˇ c ˇ ig.t/ ˇ 6 n ˇ e dtˇ 1 ; ˇ ˇ  à n ˇI ˇ inf g.n/.t/ t I j j 2 where cn is a positive constant that just depends on n. We apply this result to the function g.t/ ‚.t/ kt. Since D C n > 2, we have inf g.n/.t/ inf ‚.n/.t/ : t I j j D t I j j 2 2 Thus, for all k Z, 2 ˇ ˇ 1 ˇZ ˇ c ˇ i.‚.t/ kt/ ˇ 6 n ' I .k/ ˇ e C dtˇ 1 : j N j D 2 ˇ ˇ  à n ˇI ˇ 2 inf ‚.n/.t/ t I j j 2 In short, this means that cn ' I 6 : `1.Z/ 1 k N k  à n b 2 inf ‚.n/.t/ t I j j 2 At the same time, 1  I à 2 ' I 2 ' I 2 j j : k Nbk` .Z/ D k N k` .Z/ D 2 p 2 Now, we can interpolate ` between ` and `1 [? , Chapter 6] to obtain ' I A . / ' I `p . / k N k p T D k N k Z 1 2 2 b6 p p ' I ` . / ' I 2 k N k 1 Z k N k` .Z/ 2 0 11 p 1 B cn C  I à p 6 B C : Bb 1 C j j @  à n A 2 2 inf ‚.n/.t/ b t I j b j 2 If we plug this estimation into (27) we obtain the required result. Multipliers of sequence spaces 97 Proof of Lemma 15.4. Assume that dE .I / > 0 (otherwise the result is trivial). Hence, there is an arc J I such  that J E and J > dE .I /=2. Moreover, we may also assume that dist.J; E/ > 0. We consider a copy of J \ D; j j on R in the following way: let be an interval on R such that < 2 and j j J eit t : D f W 2 g We can write S.eit / ei‚.t/, where ‚ is a real-valued 2-periodic function and, for t , it is given by D 2 2 Z Âs t à ‚.t/ cot d.eis/: D 2 0 Since dist.J; E/ > 0, the function ‚ is infinitely differentiable on and the operator @=@t can commute with the integral. Since @3 Âs t à 3 cos2..s t/=2/ sin2..s t/=2/ cot C @t 3 2 D 4 sin4..s t/=2/ 1 4 > ; 4 sin4..s t/=2/ D eis eit 4 j j we see that Z 4 Z 4 4.I / ‚.3/.t/ > d.eis/ > d.eis/ > : eis eit 4 eis eit 4 I 4 T j j I j j j j i‚ By Proposition 9.1, the assumption S Mp is equivalent to e Mp.T/. Therefore, applying Proposition 15.5 2 2 for the third derivative of ‚, we obtain 4.I / c 6 : I 4 3 j j j j Since J > dE .I /=2, the result follows. j j D j j Now we have all the required tools to establish an important result which shows that the condition E 0 alone is j j D not enough to ensure that S Mp, for some p .1; / 2 . The set E is the boundary spectrum of S. 2 2 1 n f g Theorem 15.6 (Lebedev [15]). Let S be a singular inner function and let E be its boundary spectrum. If Eth E, ¤ then S Mp for any p with p .1; / 2 . 62 2 1 n f g Proof. Suppose that, to the contrary, there is a p .1; / 2 for which S Mp. Then, by Lemma 15.4, for any 2 1 n f g 2 arc I T, we have  I 4 6 .I / c j3 j ; dE .I / where c does not depend on I . Let us write this inequality as 3 ÂdE .I /à I 6 c j j : I .I / j j Since is singular, for -almost all T, we have 2 .I / I ! 1 j j as I contains and shrinks to this point. Therefore, at all such points, dE .I / 0; I ! j j th or equivalently, they are points of thickness for E. Hence, we can surely say .T E / 0. As E is closed and n D Eth E, we must have Eth E, which contradicts our assumption.  D 98 R. Cheng et al. To effectively use Theorem 15.6, let us introduce the concept of porous sets. We say that E T is a porous set if  there exists a constant c > 0 such that for all arcs I T  dE .I / > c I : j j For example, a singleton is a porous set. But, there are more sophisticated constructions. Clearly, if E is porous, then Eth . Hence, we immediately deduce the following corollary. D; Corollary 15.7 (Lebedev [15]). Let S be a singular inner function whose spectrum is a porous set. Then S Mp 62 for any p .1; / 2 . 2 1 n f g In particular,  1 z à sa.z/ exp a C ; a > 0; (28) D 1 z is not in Mp, for any p .1; / 2 . Using special techniques from the theory of Bessel functions, this particular 2 1 n f g result was first obtained by Verbitskii [23]. However, the original proof of Verbitskii has interesting pieces worth mentioning. Bessel functions are an important topic in the theory of special functions [8–10]. A Bessel function of the first kind and order is defined by n 1 Á2n X . 1/ x C J.x/ : WD nŠ.n 1/ 2 n 0 D C C Fix the constant > 0 and define the kernel 1 k.x/ J1. px/; x > 0: WD px From here we can define the integral operator Z1 .Kf /.x/ k.t/f .t x/ dt WD 0 p on L .RC/. Using special properties of Bessel functions, we can show that K is unbounded whenever p .2; 4/. 2 To see this, consider =2 f .x/ x J.ˇpx/; x > 0; WD where the parameters and ˇ will be determined soon. If 1 1 > 0 and . /p 1 < 0; C p 2 4 C p then f L .RC/. Moreover, one of the Sonine relations for Bessel functions [9] says that 2 ˇ =2 2 2 1=2 .Kf /.x/ x J . ˇ / px ; x > 0: D . 2 ˇ2/=2 Therefore, Z1 ˇ =2 2 2 1=2 ˇp ˇx J . ˇ / px ˇ dx p p Kf p ˇ k kL .R/ 0 p p 2 2 p=2 f Lp . / D . ˇ / Z1 k k R ˇ =2 2 2 1=2 ˇp ˇx J . ˇ / px ˇ dx 0 Z1 ˇ =2 ˇp ˇx J px ˇ dx 2.p 1/ ˇ C 0 p 2 2 p 1 : D . ˇ / C Z1 ˇ =2 ˇp ˇx J px ˇ dx 0 Multipliers of sequence spaces 99 The above estimate shows that p Kf p L .R/ k kp f p ! 1 k kL .R/ p as ˇ tends to . Hence, K is an unbounded operator on L .RC/ when p .2; 4/. Therefore, 2 bk Mp.R/; 62 where Mp.R/ is the class of Fourier integrals that is defined similarly to Mp.T/ [12]. However, the Fourier transform of the kernel k is 2 2i˛=x bk.x/ .1 e /; D where ˛ 2=8 whence D 2i˛=x e Mp.R/: 62 This assertion is equivalent to i˛= tan.x=2/ e Mp.R/: 62 The advantage of the latter is that it is a 2-periodic function on R. Thus in light of de Leeuw’s theorem [6], the atomic inner function given by the formula (28) is not a multiplier on Mp. Our second theorem has the same flavor and it leads to interesting corollaries which are easy to verify. Theorem 15.8 (Lebedev [15]). Let S be a singular inner function, and let E be its spectrum. Suppose that, for each " > 0, there are at most countably many arcs In such that [ E In  n>1 and 4 X In j3 j < ": (29) n>1 dE .In/ Then S Mp for any p .1; / 2 . 62 2 1 n f g Proof. Suppose that, to the contrary, there is a p .1; / 2 for which S Mp. Therefore, for each " > 0, there 2 1 n f g 2 is a covering of E by arcs In such that (29) holds. In the light of Lemma 15.4, we deduce that I 4 6 X 6 X n 6 .E/ .In/ c j3 j c": n>1 n>1 dE .In/ Since " is arbitrary, we must have 0 which is a contradiction. D The success of Theorem 29 lies on the fact that a covering with the property 29 forces the -size of the set to be zero. Hence, if the covering holds just for a smaller subset F E for which .F / > 0, then the conclusion of theorem  is still valid. The condition of Theorem 29 is rather difficult to verify. However, we can establish a link between this result and the length of complementary arcs of E which is way easier to handle. Corollary 15.9 (Lebedev [15]). Let E T be a closed subset of Lebesgue measure zero. Let Jn denote the arcs  that are complementary to E and ordered such that Jn > Jn 1 ; n > 1: j j j C j Assume that 1 X lim inf 3=4 Jn 0: (30) N JN j j D !1 j j n>N If S is a singular inner function whose spectrum is contained in E, then S Mp for any p .1; / 2 . 62 2 1 n f g 100 R. Cheng et al. Proof. Fix N . By removing J1;:::;JN from T, we obtain N arcs which we call K1;:::;KN . Let In be the arc of size 3 Kn and concentric with Kn. Then j j N [ E In  n 1 D and dE .In/ > min Kn ; JN : fj j j jg Note that, as an extreme case, it is possible that Jn is a singleton. In this case, to avoid certain technical difficulties, we may replace it with a small arc, e.g., of size 2 JN . Then j j N 4 4 4 X In X In X In j j 6 j j j j d 3 .I / d 3 .I / C d 3 .I / n 1 E n 16n6N E n 16n6N E n D Kn 6 JN Kn > JN j j j j j j j j .3 K /4 .3 K /4 6 X n X n j j3 j j3 Kn C JN 16n6N j j 16n6N j j Kn 6 JN Kn > JN j j j j j j j j 4 N N ! 81 6 X X 81 Kn 3 Kn j j C JN j j n 1 n 1 D j j D !4 X 81 X 81 Jn 3 Jn : D j j C JN j j n>N j j n>N By hypothesis, the right hand side can be made arbitrarily small. Therefore, the conditions of Theorem 15.8 are fulfilled and S cannot be in any Mp for any p .1; / 2 . 2 1 n f g It is easy to see that if X Jn < ; j j 1 n>1 then (30) holds. We provide further classes below. To obtain a similar useful corollary, we need a well-known concept. Given E T, its "-neighborhood is  E" T dist.; E/ < " : WD f 2 W g To measure the distance, we might use either the arc length or the Euclidian metric. The choice is irrelevant for the following result. Corollary 15.10 (Lebedev [15]). Let E T be such that  E" lim j j 0: (31) " 0 "3=4 D ! If S is a singular inner function whose spectrum is contained in E, then S Mp for any p .1; / 2 . 62 2 1 n f g Proof. Fix " > 0. Without loss of generality, we assume that E" T. Otherwise, we choose a smaller ". The set E" ¤ is a disjoint union of a finite number of arcs, e.g., I1;:::;IN . Certainly, for each arc, dE .In/ > ": Therefore, for the covering In of E, we have 4 N 4 N N ! 4 X In 1 X 4 1 X E" j j 6 In 6 In j j : 3 "3 j j "3 j j D "3 n 1 dE .In/ n 1 n 1 D D D By assumption, the right hand side can be made arbitrarily small and thus the conditions of Theorem 15.8 are fulfilled. Hence, S Mp for any p with p 2, 1 6 p 6 . 62 ¤ 1 Multipliers of sequence spaces 101 We end this section by showing that a wide range of generalized Cantor sets fall into the family of sets described above. Let .n/n>0 be a sequence in the interval .0; 1/. Let I be a closed arc on T of size 0 I . The generalized D j j Cantor set is constructed as follows: Step 1: from I we remove the concentric open arc of length 00. As so, we obtain two arcs of length 1 1 .1 0/0: D 2 Step 2: from each of the remaining two arcs, we remove an open concentric arc of length 11. As so, we obtain four arcs of length 1 2 .1 1/1: D 2 Step n: we continue the above process and in the n-th step we would have 2n intervals of length .1 n 1/n 1 .1 n 1/ .1 0/0 n : D 2 D 2n The remaining set, which can be written as the intersection of the union of the above 2n arcs, is called the generalized Cantor set. It is rather straightforward to see that if N 1 1 Y lim .1 n/ 0 N 3 D !1 N n 1 D then the conditions of Theorem 15.8 are fulfilled. Hence, S is any singular inner function whose spectrum is contained in E then S Mp for any p .1; / 2 . 62 2 1 n f g 16 Orthogonality The notion of Birkhoff-James orthogonality [2, 13] extends the concept of orthogonality from an inner product space to a more general normed linear space. Let x and y be vectors belonging to a normed linear space X . We say that x is orthogonal to y in the Birkhoff-James sense if x ˇy X > x X (32) k C k k k for all scalars ˇ. In this situation we write x X y. It is straightforward to show that when X is a Hilbert space, then ? p x y is equivalent to x X y. The relation X is generally neither symmetric nor linear. When X ` , let us ? ? ? D write p in place of the more cumbersome `p . Of particular importance here is the following explicit criterion for ? ? the relation p when p .1; /. ? 2 1 Theorem 16.1 (James [13]). X1 p 2 a p b ak akbk 0; (33) ? ” j j D k 0 D where any occurrence of “ 0 p 20” in the sum above is interpreted as zero. j j Borrowing from (33) we define, for a complex number ˛ rei , and any s > 0, the quantity D ˛ s .rei / s rse i : (34) h i D h i WD It is easy to verify that for any complex numbers ˛ and ˇ, real exponent s > 0, and integer n > 0, we have .˛ˇ/ s ˛ s ˇ s h i D h i h i ˛ s ˛ s j h ij D j j 102 R. Cheng et al. ˛ s ˛ ˛ s 1 h i D j j C .˛ s /n .˛n/ s h i D h i .˛ p 1 / q 1 ˛: h i h i D In light of the definition (34), for a .ak/k>0, let us write D p 1 p 1 a .ah i/k>0: (35) h i WD k If a `p, it is easy to see that a p 1 `q and thus from (2) and (33), 2 h i 2 p 1 a p b .b; a / 0: (36) ? ” h i D Note that p is therefore linear in its second argument, when p .1; /, and it then makes sense to speak of a ? 2 1 vector being orthogonal to a subspace of `p. p p p p Due to the isometry between ` and `A, we can pass the Birkhoff-James orthogonality from ` to `A. More explicitly, if X1 k X1 k a.z/ akz ; and b.z/ bkz D D k 0 k 0 D D p are in `A, then a p b a p b: ? ” ? Similarly, we define p 1 X1 p 1 k a .z/ ah iz : h i D k k 0 D Note that a p 1 `q and (36) is rewritten as h i 2 A p 1 a p b .b; a / 0: (37) ? ” h i D 17 An application to zeros of analytic functions In this section we discuss an application of multipliers and Birkhoff-James orthogonality to estimating zeros of polynomials from [20]. We first introduce a special function that enables us to connect an analytic function to an orthogonality condition. For p .1; / and w D 0 , define 2 1 2 n f g 1 z=w Bp;w.z/ : WD 1 w q 1 z h i q 1 q 1 2 1 Since wh i w < 1, the function Bp;w is analytic in D. When p 2 observe that wh i w and so j j D j j D D 1 w z B2;w ; (38) D w 1 wz i 1 which is just a constant multiple of a Blaschke factor. Using the fact that B2;w.e / w for all , we see that j j D j j 2 2 Z Z i k i d 1 ik d B2;w.e /S B2;w.e / e 0; k > 1: 2 D w 2 2 D 0 j j 0 k Thus B2;w 2 S B2;w for all k > 1. It turns out that something analogous holds when p .1; /. We refer the ? 2 1 reader to [20] for the proof. Lemma 17.1. For each p .1; / and w D 0 we have 2 1 2 n f g Multipliers of sequence spaces 103 p (i) Bp;w p Bp;wf for all f `A with f .0/ 0; ? q2p 1 1=p D h .1 w / i (ii) Bp;w p 1 j j : k k D C w p j j p 2 It could be said that Bp;w plays a role in `A analogous to that of a Blaschke factor in the Hardy space H . However, here the situation is more complicated. See [5] for an exploration of this idea. p Here is our application of `A multipliers to obtain another proof of a set of classical bounds for the zeros of an analytic function. These bounds are tied to the well-known estimates for polynomial roots by Cauchy, Lagrange, and others (see, for example, [19]). Yet another proof of this result appears in [20], along with some extensions, based on using Birkhoff-James orthogonality more directly. Theorem 17.2. Suppose that X1 k f .z/ akz D k 0 D is analytic in D, and a0 0. If w D is a zero of f , then ¤ 2 Ä 1 ˇa1 ˇ ˇa2 ˇ ˇa3 ˇ w > ˇ ˇ ˇ ˇ ˇ ˇ (39) j j ˇa0 ˇ C ˇa0 ˇ C ˇa0 ˇ C Ä 1 nˇa1 ˇ ˇa2 ˇ ˇa3 ˇ o w > 1 sup ˇ ˇ; ˇ ˇ; ˇ ˇ; (40) j j C ˇa0 ˇ ˇa0 ˇ ˇa0 ˇ and Ä 1=q ˇa1 ˇp ˇa2 ˇp ˇa3 ˇp Áq=p w > 1 ˇ ˇ ˇ ˇ ˇ ˇ (41) j j C ˇa0 ˇ C ˇa0 ˇ C ˇa0 ˇ C for all p .1; /. 2 1 Proof. Fix p .1; / and assume that f `p. Otherwise the right hand side is zero and the inequality is 2 1 2 A automatically true. Recall that _ Œf zkf k > 0 D f W g p and note that SŒf is a (closed) subspace of `A since S is an isometry. Write fe for the metric projection of f onto p p SŒf . Since ` is uniformly convex [4], this metric projection is the unique function fe ` satisfying A 2 A inf f g p g SŒf f fe p: (42) fk k W 2 g D k k Let f .z/ f1.z/ w.Qwf /.z/ WD 1 z=w D and note, by hypothesis, that f1 analytic in D and f1.0/ f .0/ a0: (43) D D p Furthermore, by Proposition 6.1, we have f1 ` . If P denotes the analytic polynomials, we have 2 A f p > f fe p (by (42)) (44) k k k k inf f zqf p q P D fk C k W 2 g z Á z Á inf f1 1 zq 1 f1 q P D w C w p W 2 z Á inf f1 1 .1 zq/ q P D w C p W 2 z Á inf f1 1 Q Q P; Q.0/ 1 : (45) D w p W 2 D 104 R. Cheng et al. For any Q P with Q.0/ 1 we have the identity 2 D q 1 z Á f1Bp;w.1 w z/Q f1 1 Q: (46) h i D w Furthermore, if Q X1 k dkz 1 w q 1 z D h i k 0 D and n X k Qn dkz ; D k 0 D then Qn P with Qn.0/ 1 and by Corollary 12.2, we get 2 D Q Qn q 1 0; 1 w z Mp ! h i From here we see that z Á z Q 6 Qnf1 1 f1Bp;wQ 1 Qn q 1 f1 p w p w Mp 1 w z Mp k k h i which goes to zero as n . Combine this with (46) to see that the sets ! 1 z f1Bp;wQ Q P; Q.0/ 1 and f1.1 /Q Q P; Q.0/ 1 f W 2 D g f w W 2 D g p have the same closure in `A. Thus (45) is equal to inf f1Bp;wQ p Q P; Q.0/ 1 : (47) fk k W 2 D g In a somewhat similar way, let Q P with Q.0/ 1. If 2 D X1 k f1Q ckz D k 0 D and n 1 X k Qn ckz ; D a0 k 0 D then Qn P with Qn.0/ 1 (see (43)) and 2 D 0 11=p X1 p a0Qn f1Q p @ ck A 0: k k D j j ! k n 1 D C Thus a0Bp;wQn f1Bp;wQ p 6 Bp;w a0Qn f1Q p 0: k k k kMp k k ! This says that the closure of a0Bp;wQ Q P; Q.0/ 1 f W 2 D g p in `A is contained in the closure of f1Bp;wQ Q P; Q.0/ 1 : f W 2 D g From this containment of closures, we see that (47) is bounded below by a0 inf Bp;wQ p Q P; Q.0/ 1 : (48) j j fk k W 2 D g Using the fact that Q.0/ 1, write D Bp;wQ Bp;w .Q Q.0//Bp;w D C Multipliers of sequence spaces 105 and use Lemma 17.1 and (32) to see that (48) is bounded below by a0 Bp;w p: j j k k By Lemma 17.1, the above is equal to q p 1 1=p h .1 w / i a0 1 j j : j j C w p j j Following this all the way back to (44) yields the inequality q p 1 1=p h .1 w / i f p > a0 1 j j k k j j C w p j j from which it follows .1 w q/p 1 ˇa ˇp ˇa ˇp ˇa ˇp 6 ˇ 1 ˇ ˇ 2 ˇ ˇ 3 ˇ j jp : w ˇa0 ˇ C ˇa0 ˇ C ˇa0 ˇ C j j Writing nˇa1 ˇp ˇa2 ˇp ˇa3 ˇp o1=p M ˇ ˇ ˇ ˇ ˇ ˇ ; WD ˇa0 ˇ C ˇa0 ˇ C ˇa0 ˇ C we have .1 w q/p 1 j j 6 M p w p j j .1 w q/p 1 6 w pM p j j j j .1 w q/1=q 6 w M j j j j .1 w q/ 6 w qM q j j j j 1 6 w : .M q 1/1=q j j C This proves (41). The bounds in (39) and (40) are obtained by taking the limits p 1 and p , respectively. ! ! 1 18 Coefficient estimates p We know from Corollary 8.4 that if h Mp then in fact h ` and 2 2 A h p 6 h M : (49) k k k k p p 1 If we use the hh i idea via Birkhoff-James orthogonality, we can sharpen this inequality considerably. Proposition 18.1. Suppose that p .1; / and h Mp. Then 2 1 2 q p 1 h p p p Á h h > a0 a1 a2 (50) k kp k kMp j j C j j C j j C q ˇ p 1 p 1 p 1 ˇ ˇah ia0 ah ia1 ah ia2 ˇ C ˇ 1 C 2 C 3 C ˇ q ˇ p 1 p 1 p 1 ˇ ˇah ia0 ah ia1 ah ia2 ˇ C ˇ 2 C 3 C 4 C ˇ q ˇ p 1 p 1 p 1 ˇ ˇah ia0 ah ia1 ah ia2 ˇ C ˇ 3 C 4 C 5 C ˇ i1=q C P k p p Proof. Let f .z/ 1 bkz belong to ` . Let us write f for the isomorph of f in the sequence space ` and D k 0 A hf for the correspondingD sequence for hf . Then for any c `q, the linear functional 2 f .hf; c/ 7! 106 R. Cheng et al. p is continuous on ` with norm no greater than h c q. In particular, we could consider the basic vectors A k kMp k k e0 .1; 0; 0; 0; : : :/; WD e1 .0; 1; 0; 0; : : :/; WD e2 .0; 0; 1; 0; : : :/; WD : : for `q, where we find that k X hf; ek ak j bj h i D j 0 D .f; .ak; ak 1; : : : ; a1; a0; 0; 0; : : ://: D Set .k/ u .ak; ak 1; : : : ; a1; a0; 0; 0; : : :/: WD q It follows that for any c .c0; c1; c2; : : :/ ` , we have WD 2 X1 .k/ .hf; c/ .f; ckh / D k 0 D If we interpret this operation X1 .k/ f .f; cku / 7! k 0 D as a bounded linear functional on `p, the Riesz Representation Theorem says that the sequence X1 .k/ cku k 0 D must belong to `q. Let us write out what that means term by term: ˇ ˇq ˇa0c0 a1c1 a2c2 ˇ ˇ C C C ˇ ˇ ˇq ˇa0c1 a1c2 a2c3 ˇ C ˇ C C C ˇ ˇ ˇq ˇa0c2 a1c3 a2c4 ˇ C ˇ C C C ˇ C q 6 h c q : (51) k kMq k k` Now it is a simple matter to substitute p 1 p 1 p 1 c .ah i; ah i; ah i; : : :/; D 0 1 2 which belongs to `q, to obtain (50). Remark 18.2. Notice that if you drop all but the first line in the right side of (50), then you obtain (49), and so Proposition 18.1 does indeed represent a dramatic sharpening of (49). Also notice that if h is Birkhoff-James p k orthogonal in `A to z h for all positive integers k, then in fact all but the first line in the right side of (50) is zero. This is the case when h Bp;w. D Multipliers of sequence spaces 107 19 Hadamard multipliers Let f and g be two analytic functions on D with Taylor series expansions X1 n X1 n f .z/ anz and g.z/ bnz : D D n 0 n 0 D D The Hadamard product of f and g is defined by X1 n .f g/.z/ anbnz : ˘ D n 0 D Since f and g are analytic on D, it is easy to verify that f g is also analytic on D. Note that the disk of convergence ˘ p might be larger, e.g., consider odd and even functions. We say that h is a Hadamard multiplier of `A if the operator p p Mh ` ` ;Mhf h f: W A ! A D ˘ is well-defined and continuous. As a rule of thumb, if the norm is defined via Taylor coefficients, e.g., Dirichlet 2 2 p space, Hardy space H , Bergman space A , and `A spaces, then determining the Hadamard multipliers if usually not a difficult task. However, if the norm is defined differently, e.g., via an integral as in the Hardy spaces H p, the p Bergman spaces A , and the harmonically weighted Dirichlet spaces D, determining the Hadamard multipliers is non-trivial. In [7] one can find a discussion of the Hadamard multipliers between various H p classes. Here is a p characterization of the Hadamard multipliers for `A. Theorem 19.1. Let p .1; /. Then the Hadamard multiplier space of `p is isometrically isomorphic to ` . 2 1 A A1 More explicitly, for each h ` , we have 2 A1 Mh h ` : k k D k k A1 Conversely, if Mh < , then h ` . k k 1 2 A1 P n P n p Proof. First suppose that h.z/ n1 0 anz `A1. Then for each f .z/ n1 0 bnz `A, we have D D 2 D D 2 1 ! p X1 p Mhf `p anbn k k A D j j n 0 D 1  à ! p X1 p 6 sup an bn > j j j j n 0 n 0 D h ` f `p : D k k A1 k k A Thus h is in fact a Hadamard multiplier and moreover, Mh 6 h ` : k k k k A1 m To show that equality holds, consider the monomials fm.z/ z , m > 0. Then fm `p 1 and D k k A D 1 ! p X1 p Mhf `p anbn am : k k A D j j D j j n 0 D Thus Mh > am m > 0; k k j j which implies the reverse inequality Mh > h ` : k k k k A1 108 R. Cheng et al. By a similar argument, we can show that if Mh < , then h ` . In fact, we must have k k 1 2 A1 Mhf `p 6 Mh f `p k k A k k k k A p for all f ` . In particular, if we apply the inequality above to fm, we obtain 2 A am 6 Mh ; m > 0: j j k k Taking supremum with respect to m implies h ` . 2 A1 References [1] Evgeny Abakumov and Alexander Borichev. Shift invariant subspaces with arbitrary indices in lp spaces. J. Funct. Anal., 188(1):1–26, 2002. [2] J. Alonso, H. Martini, and S. Wu. 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