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]).