arXiv:2007.11217v3 [math.FA] 26 Aug 2021 eog to belongs n hc scle h cu ls.Frayfunction any For class. Schur the called is which and Let Introduction 1 ossigo l one nltcfntoson functions analytic bounded all of consisting etaso eBagsRvykkernels”. Branges-Rovnyak de of nentials ∗ hspprhsbe cetdb aainMteaia ultn nw in Bulletin, Mathematical Canadian by accepted been has paper This D anterm ti hw htteepnnilo eBranges- de correspon of the of exponential part the inner the nontrivial. that if shown definite positive is strictly it theorem, main 00Mteaia ujc lsicto:Piay3H5 Second 30H45; Primary Classification: Subject Mathematical 2010 eteoe ntds ntecmlxplane complex the in disk unit open the be prtrter nue ypwr ftede the of powers by induced theory Operator nti oe egv e rpryo eBagsRvykker Branges-Rovnyak de of property new a give we note, this In rne-ona enladisapplication its and kernel Branges-Rovnyak S fadol if only and if ewrs eBagsRvykkre,pstv ent kernel definite positive kernel, Branges-Rovnyak de keywords: apr esuHg col apr 6-96 Japan 064-0916, Sapporo School, High Seishu Sapporo ainlDfneAaey ooua2988,Japan 239-8686, Yokosuka Academy, Defense National -aladdress E-mail S -aladdress E-mail = hhiKUWAHARA Shuhei { ϕ : ∈ ihoSETO Michio [email protected] H 1 ∞ − Abstract : 1 ϕ : | and − ϕ ( [email protected] 1 λ ( λz λ ) ϕ ) D ≤ | ( hn eset we Then, . z ) C ( 1 n let and , λ ϕ ∈ in D igShrcasfnto is function class Schur ding ) H } H , ∞ ∞ ti elkonthat known well is it , ihtenwttei “Expo- is title new the hich ona enlis kernel Rovnyak eteBnc algebra Banach the be es sthe As nels. r 15B48 ary ∗ (1) ϕ 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 −| | . k k H 1 λ 2 −| | Thus we have (3).
3 Unbounded multipliers
We shall investigate into unbounded multipliers of exp (ϕ). Ht Lemma 3.1. Let ψ be a function in (ϕ). Then, for any function f in (ϕ)n, ψf Ht Ht belongs to (ϕ)n+1. Ht Proof. We define bounded linear operator τψ as follows: n n+1 τ : (ϕ)⊗ (ϕ)⊗ , F ψ F. ψ Ht → Ht 7→ ⊗ Then, the following diagram commutes:
n τψ n+1 (ϕ)⊗ (ϕ)⊗ Ht −−−→ Ht
∆n ∆n+1
n n+1 t(ϕ) t(ϕ) , H −−−−−−→Mψ H (ϕ)n H y | t y
where Mψ denotes the multiplication operator with symbol ψ. This concludes the proof.
Theorem 3.2. Let ψ be a function in t(ϕ). Then, the multiplication operator Mψ is a densely defined closable linear operator inH exp (ϕ). Ht n Proof. Let F = (f , f ,...,f , 0 ...)⊤ be a vector having finite support in ∞ (ϕ) . 0 1 N ⊕n=0Ht We set ΓF = f. Then,
N 1 N 1 N 1 N+1 1 ψf = ψ fn = ψfn = (n + 1)ψfn = nψfn 1, n! n! (n + 1)! n! − n=0 n=0 n=0 n=1 X X X X n where we note that nψfn 1 belongs to t(ϕ) by Lemma 3.1. Hence, setting − H G = (0, ψf0, 2ψf1,..., (N + 1)ψfN , 0,...)⊤,
n G belongs to n∞=0 t(ϕ) and ΓG = ψf, that is, ψf belongs to exp t(ϕ). Therefore, Mψ is a densely defined⊕ H linear operator in exp (ϕ). Moreover, it is easyH to see that M is Ht ψ closable.
6 Corollary 3.3. Let ψ be a function in t(ϕ). Then the adjoint operator Mψ∗ of Mψ is a densely defined closed linear operator inHexp (ϕ), and every exp tkϕ is an eigenfunction Ht λ of Mψ∗. More precisely, ϕ ϕ Mψ∗ exp tkλ = ψ(λ) exp tkλ .
4 Main results
Let X be a set. A function k on X X is called a strictly positive definite kernel on X if k(x, y)= k(y, x) for any x and y in ×X and
n
cicjk(xj, xi) > 0 i,j=1 X n for any n in N, any (c1,...,cn)⊤ in C 0 and any n distinct points x1,...,xn in X. For example, it is well known that \{ } k(z,λ) = exp(λz) is a strictly positive definite kernel on C. In fact, this is the reproducing kernel of the Segal-Bargmann space. Now, we note that if ϕ = z2 then
1 1 ϕ(λ)ϕ(z) e− exp − = exp(λz). 1 λz ! − Motivated by this observation, we shall give new examples of strictly positive definite kernels. Let ϕ be a function in . For the canonical factorization ϕ = αzN BSF of ϕ (see Section 5 in Chapter II in GarnettS [7]), where α is a unimodular constant, B is a Blaschke product consisting of nonzero zero points, S is a singular inner function and F is an outer function, we consider three conditions (C1) N 2, (C2) B is nontrivial and (C3) S is nontrivial. We need the following lemma. ≥
Lemma 4.1. Let λ1,...,λn be n distinct points in D. Suppose one of (C1), (C2) and (C3). Then there exists a function ψ in (ϕ) such that ψ(λ ) = ψ(λ ) (i = j). Ht i 6 j 6
Proof. Since t(ϕ) = (ϕ) as vector spaces, it suffices to show the statement for (ϕ). First, we assumeH (C1).H Then, since ϕ/z is in by the Schwarz lemma and (ϕ/z)(0)H = 0, S we have (I T T ∗)z = z T T ∗ T ∗z = z T T ∗ 1= z, − ϕ ϕ − ϕ ϕ/z z − ϕ ϕ/z 2 where Tϕ denotes the Toeplitz operator with symbol ϕ on the Hardy space H over D. Hence z belongs to (ϕ), and we may take ψ = z. H Secondly, we assume (C2). Let µ be a nonzero zero point of ϕ. Then, we have
ϕ k =(I T T ∗)k = k . µ − ϕ ϕ µ µ 7 1 1 Hence (1 µz)− belongs to (ϕ), and we may take ψ = (1 µz)− . Thirdly,− we assume (C3).H Then, observe that there exists− a point θ [0, 2π) such that √ 1θ ∈ ϕ(re − ) 0 as r 1 (see Theorem 6.2 in [7]). Without loss of generality, we may assume that θ = 0.→ Setting ↑
ϕ ϕ δij(r)= kr (λi) kr (λj) and δ(r) = min δij(r) (0 it suffices to show that δ(r) > 0 for any r sufficiently close to 1. For any distinct λi and 1 λ , we suppose that, for any m in N, there exists a real number r (1 m− , 1) such j m ∈ − that δij(rm) = 0. Then, we have 1 1 1 ϕ(rm)ϕ(λi) 1 ϕ(rm)ϕ(λj) = lim − − = lim δij(rm)=0. 1 λi − 1 λj m 1 rmλi − 1 rmλj m − − →∞ − − →∞ This concludes that λ = λ , however, which contradicts the assumption λ = λ . Hence i j i 6 j there exists rij (0, 1) such that δij(r) > 0 for any r (rij, 1). Setting R = max1 i 8 Further, by Lemma 4.1, there exists a function ψ in (ϕ) such that Ht (ψ(λ ) ψ(λ )) =0. i − j 6 1 i 1 1 ··· ψ(λ1) ψ(λn) . ···. . . . . n 1 n 1 ψ(λ ) − ψ(λ ) − 1 n ··· is nonsingular. Therefore, we have that ϕ c1 exp tkλ ϕ1 c2 exp tkλ . 2 = 0. . ϕ cn exp tk λn This concludes that c = = c = 0. 1 ··· n The well-known fact mentioned at the beginning of this section is included in Theorem 4.2. Corollary 4.3. The kernel function k(z,λ) = exp(λz) is strictly positive definite on C. Proof. For any n distinct points λ1,...λn in C, we set R = max1 j n λj + 1. Then ≤ ≤ 2 | | 2 λ1/R,...λn/R are in D. Hence, by Theorem 4.2 in the case where ϕ = z and t = R , we have n n R2 2 cicj exp(λiλj)= e− cicj exp(R + λiλj) i,j=1 i,j=1 X Xn R2 2 = e− cicj exp(R (1 + (λi/R)(λj/R))) i,j=1 X n R2 2 1 ϕ(λi/R)ϕ(λj/R) = e− cicj exp R − > 0 i,j=1 1 (λi/R)(λj/R) ! X − n for any (c ,...,c )⊤ in C 0 . 1 n \{ } 9 Although the next result is just a simple consequence of Theorem 4.2, from the view- point of the theory of model spaces (see Garcia-Mashreghi-Ross [8]), it will be worth while mentioning it as a theorem. Theorem 4.4. Let ϕ be an inner function. If ϕ is neither a constant nor eiθz, then the kernel 1 ϕ(λ)ϕ(z) kt(z,λ) = exp t − (t> 0) 1 λz ! − is strictly positive definite. Further, with help of the theory of sub-Hardy Hilbert spaces, we have Theorem 4.5. Let ϕ be a function in . If ϕ is a nonextreme point of the closed unit ball S in H∞, then the kernel 1 ϕ(λ)ϕ(z) kt(z,λ) = exp t − (t> 0) 1 λz − ! is strictly positive definite. Proof. By Theorem (IV-3) in [15] (or Theorem 23.13 in [6]), the polynomials are dense in (ϕ). Hence z belongs to t(ϕ). Then, setting ψ = z, the proof of Theorem 4.2 applies Hto this case. H References [1] A. Abkar and B. Jafarzadeh, Weighted sub-Bergman Hilbert spaces in the unit disk. Czechoslovak Math. J. 60(135) (2010), no. 2, 435–443. [2] J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, 44. American Mathematical Society, Providence, RI, 2002. [3] J. A. Ball and V. Bolotnikov, Interpolation in sub-Bergman spaces. Advances in structured operator theory and related areas, 17–39, Oper. Theory Adv. Appl., 237, Birkh¨auser/Springer, Basel, 2013. [4] J. A. Ball and V. Bolotnikov, de Branges-Rovnyak spaces: basics and theory, arXiv:1405.2980. [5] C. 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