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Operator Theory Induced by Powers of the De Branges-Rovnyak Kernel and Its Application

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Operator Theory Induced by Powers of the De Branges-Rovnyak Kernel and Its Application Operator theory induced by powers of the de Branges-Rovnyak kernel and its application∗ Shuhei KUWAHARA Sapporo Seishu High School, Sapporo 064-0916, Japan E-mail address: [email protected] and Michio SETO National Defense Academy, Yokosuka 239-8686, Japan E-mail address: [email protected] Abstract In this note, we give a new property of de Branges-Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges-Rovnyak kernel is strictly positive definite if the inner part of the corresponding Schur class function is nontrivial. 2010 Mathematical Subject Classification: Primary 30H45; Secondary 15B48 keywords: de Branges-Rovnyak kernel, positive definite kernel arXiv:2007.11217v3 [math.FA] 26 Aug 2021 1 Introduction Let D be the open unit disk in the complex plane C, and let H∞ be the Banach algebra consisting of all bounded analytic functions on D. Then, we set = ϕ H∞ : ϕ(λ) 1 (λ D) , S { ∈ | | ≤ ∈ } and which is called the Schur class. For any function ϕ in H∞, it is well known that ϕ belongs to if and only if S 1 ϕ(λ)ϕ(z) − (1) 1 λz − ∗This paper has been accepted by Canadian Mathematical Bulletin, in which the new title is “Expo- nentials of de Branges-Rovnyak kernels”. 1 is positive semi-definite. This equivalence relation based on the properties of the Szeg¨o kernel is crucial in the operator theory on the Hardy space over D, in particular, theories of Pick interpolation, de Branges-Rovnyak spaces and sub-Hardy Hilbert spaces (see Agler- McCarthy [2], Ball-Bolotnikov [4], Fricain-Mashreghi [6] and Sarason [15]). The kernel (1) is called the de Branges-Rovnyak kernel. Before introducing our study, we should mention that not only the original de Branges- Rovnyak kernel but also its variants have been studied by a number of authors. For example, Zhu [17, 18] initiated the study on the kernel 1 ϕ(λ)ϕ(z) − (2) (1 λz)2 − in the Bergman space over D. The reproducing kernel Hilbert space induced by the kernel (2) is called a sub-Bergman Hilbert space (see also Abkar-Jafarzadeh [1], Ball- Bolotnikov [3], Chu [5], Nowak-Rososzczuk [12] and Sultanic [16]). Further, powers of the de Branges-Rovnyak kernel n 1 ϕ(λ)ϕ(z) − (n N) (3) 1 λz ! ∈ − are naturally obtained from the theory of hereditary functional calculus for weighted Bergman spaces on D (see Example 14.48 in [2] for the case where n = 2) and have appeared also in p. 3672 of Jury [9]. Now, the purpose of this paper is to study the structure of the kernel 1 ϕ(λ)ϕ(z) exp t − (t> 0). (4) 1 λz − ! Note that our kernel (4) is obtained by binding all kernels in (3) together. Thus, we expect that new properties of the de Branges-Rovnyak kernel (1) are drawn out from our kernel (4). In fact, as the main result, we will show that the exponential of the de Branges-Rovnyak kernel is strictly positive definite if the inner part of ϕ is nontrivial. Here, we shall give some remarks on strictly positive definite kernels. In general, it is not difficult to construct positive semi-definite kernels. On the other hand, for strictly positive definite kernels, nontrivial methods depending on each case are often needed (for example, see Micchelli [10]). Moreover, it might be worth while mentioning that strictly positive definite kernels have received attention in machine learning (see Rasmussen-Williams [14]). This paper is organized as follows. In Section 2, basic properties of the reproducing kernel Hilbert space exp (ϕ) constructed from our kernel (4) are given. In Section 3, Ht unbounded multipliers on exp t(ϕ) are introduced and studied. In Section 4, main results are given. H The problem discussed in this paper was obtained in conversation about machine learn- ing with Professor Kohtaro Watanabe (National Defense Academy). The authors would like to express gratitude to him. This research was supported by JSPS KAKENHI Grant Number 20K03646. 2 2 Preliminaries For t> 0, let (ϕ) denote the reproducing kernel Hilbert space with kernel Ht 1 ϕ(λ)ϕ(z) tkϕ(z,λ)= t − (ϕ ), 1 λz ∈ S − and we will use notations tkϕ(z)= tkϕ(z,λ) and (ϕ)= (ϕ). Then, since λ H H1 ϕ ϕ ϕ 1 ϕ ϕ tkλ , tkz t(ϕ) = tk (z,λ)= t− tkλ , tkz (ϕ), h iH h iH the trivial linear mapping f f from (ϕ) onto (ϕ) is bounded and invertible. Par- 7→ H Ht ticularly, t(ϕ) = (ϕ) as vector spaces. In this section, we construct the exponential of (ϕ)H and giveH its basic properties. The contents of this section are well known to Ht specialists. For example, see Exercise (k) in p. 320 of Nikolski [11] and Chapter 7 in Paulsen-Raghupathi [13]. However, we give the details for the sake of readers. 2.1 Construction of exp Ht(ϕ) Let (ϕ)n be the reproducing kernel Hilbert space obtained by the pull-back construction Ht with the n-fold tensor product space n (ϕ)⊗ = (ϕ) (ϕ) Ht Ht ⊗···⊗Ht and the n-dimensional diagonal map ∆ : D Dn, λ (λ,...,λ) n → → ϕ n (for the pull-back construction, see Theorem 5.7 in [13]). We note that (tkλ )⊗ ∆n = ϕ n n n ◦ (tkλ ) is the reproducing kernel of t(ϕ) . Let n∞=0 t(ϕ) denote the Hilbert space with the inner product H ⊕ H ∞ 1 ∞ n n (f0, f1,...)⊤, (g0,g1,...)⊤ t(ϕ) = fn,gn t(ϕ) , h i⊕n=0H n!h iH n=0 X where we set (ϕ)0 = C. Moreover, we define linear map Γ as follows: Ht f0 f0 ∞ 1 n Γ: f1 fn f1 ∞ t(ϕ) . . 7→ n! . ∈ ⊕n=0H . n=0 . X Proposition 2.1. The following statements hold: n (i) Γ is a map from ∞ (ϕ) to Hol(D). ⊕n=0Ht (ii) ker Γ is closed. 3 n Proof. For any F =(f , f ,...)⊤ in ∞ (ϕ) , we have 0 1 ⊕n=0Ht m 1 m 1 f (λ) f (λ) ℓ! ℓ ≤ ℓ! ℓ ℓ=n+1 ℓ=n+1 X X m 1 ϕ ℓ ℓ ℓ fℓ t(ϕ) (tkλ ) t(ϕ) ≤ ℓ!k kH k kH ℓ=Xn+1 m 1/2 m 1/2 1 2 1 ϕ ℓ 2 ℓ ℓ fℓ t(ϕ) (tkλ ) t(ϕ) ≤ ℓ!k kH ! ℓ!k kH ! ℓ=Xn+1 ℓ=Xn+1 m 1/2 m 1/2 1 2 1 ϕ 2ℓ ℓ = fk t(ϕ) tkλ t(ϕ) . ℓ!k kH ! ℓ!k kH ! ℓ=Xn+1 ℓ=Xn+1 Hence, ∞ 1 f (λ) n! n n=0 X converges uniformly on any compact subset of D. This concludes (1). Next, suppose that (ℓ) (ℓ) n F = (f , f ,...)⊤ belongs to ker Γ and F converges to F in ∞ (ϕ) . Then, for ℓ 0 1 ℓ ⊕n=0Ht sufficiently large L, we have ∞ 1 (ℓ) 2 2 n ∞ n fn fn t(ϕ) = Fℓ F t(ϕ) < 1 (ℓ L). n!k − kH k − k⊕n=0H ≥ n=0 X (ℓ) n Hence we have fn fn t(ϕ) < √n! for every n 0 if ℓ L. It follows from this H inequality that,k for any− ℓ k L, ≥ ≥ ≥ 1 (ℓ) 1 (ℓ) ϕ n n n fn (λ) fn t(ϕ) (tkλ ) t(ϕ) n! ≤ n!k kH k kH ϕ n n √ ( fn t(ϕ) + n!) tkλ t(ϕ) k kH k kH ≤ n! ϕ n (M + 1) tkλ t(ϕ) k kH , ≤ √n! 2 1/2 where we set M = supn 0( fn t(ϕ)/n!) . Then, by the ratio test, ≥ k kH ϕ n ∞ (M + 1) tkλ t(ϕ) k kH √ n=0 n! X is finite. Hence, by the Lebesgue dominated convergence theorem and the assumption that (ℓ) (ℓ) Fℓ =(f0 , f1 ,...)⊤ belongs to ker Γ, we have ∞ ∞ ∞ 1 1 (ℓ) 1 (ℓ) fn(λ)= lim fn (λ) = lim fn (λ)=0. n! ℓ n! ℓ n! n=0 n=0 →∞ →∞ n=0 X X X 4 This concludes (2). By Proposition 2.1, the pull-back construction can be applied to Γ. Definition. We define exp (ϕ) as the reproducing kernel Hilbert space obtained by the Ht pull-back construction with the linear map n Γ: ∞ (ϕ) Hol(D). ⊕n=0Ht → 2.2 Basic properties of exp Ht(ϕ) We summarize basic properties of exp (ϕ). Ht Proposition 2.2. exp (ϕ) is a reproducing kernel Hilbert space consisting of holo- Ht morphic functions on D. More precisely, for any f in exp t(ϕ), there exists a vector n H (f , f ,..., )⊤ in ∞ (ϕ) such that 0 1 ⊕n=0Ht ∞ 1 f = f n! n n=0 X converges uniformly on any compact subset of D. Moreover, (i) the following norm estimate holds: ∞ 2 1 2 n f exp t(ϕ) fn t(ϕ) , k k H ≤ n!k kH n=0 X (ii) the reproducing kernel of exp (ϕ) is Ht ∞ 1 (tkϕ)n = exp tkϕ, n! λ λ n=0 X that is, ϕ f(λ)= f, exp tkλ exp t(ϕ) h i H for any λ in D, (iii) the following growth condition holds: 2 2 2 1 ϕ(λ) f(λ) f exp t(ϕ) exp t −| | | | ≤k k H 1 λ 2 −| | for any λ in D. 5 Proof. By the definition of the norm and the inner product of exp t(ϕ), we have (1) and (2). We shall show (3). By (2) and the Cauchy-Schwarz inequality,Hwe have 2 ϕ 2 f(λ) = f, exp tkλ exp t(ϕ) H | | |h 2 i | ϕ 2 f exp t(ϕ) exp tkλ exp t(ϕ) ≤k k H ·k k H 2 ϕ = f exp t(ϕ) exp tkλ (λ) k k H · 2 2 1 ϕ(λ) = f exp t(ϕ) exp t −| | .
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