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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/ .
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