Hostname: page-component-7dd5485656-hw7sx Total loading time: 0 Render date: 2025-10-30T00:46:58.973Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponentials of de Branges–Rovnyak kernels

Part of: Special matrices

Published online by Cambridge University Press:  03 June 2021

Shuhei Kuwahara  and
Michio Seto
Shuhei Kuwahara
Affiliation:
Sapporo Seishu High School, Sapporo, Japan e-mail: s.kuwahara@sapporoseishu.ed.jp
Michio Seto*
Affiliation:
Department of Mathematics, National Defense Academy, Yokosuka, Japan
*
e-mail: mseto@nda.ac.jp
Get access Rights & Permissions [Opens in a new window]

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 corresponding Schur class function is nontrivial.

Keywords

de Branges–Rovnyak kernelpositive definite kernelreproducing kernel Hilbert space

MSC classification

Secondary: 15B48: Positive matrices and their generalizations; cones of matrices

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 2 , June 2022 , pp. 447 - 455
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abkar, A. and Jafarzadeh, B., Weighted sub-Bergman Hilbert spaces in the unit disk. Czechoslovak Math. J. 60(2010), no. 2, 435–443.CrossRefGoogle Scholar
Agler, J. and McCarthy, J. E., Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, 44, American Mathematical Society, Providence, RI, 2002.CrossRefGoogle Scholar
Ball, J. A. and Bolotnikov, V., Interpolation in sub-Bergman spaces. In: Advances in structured operator theory and related areas, Oper. Theory Adv. Appl., 237, Birkhäuser/Springer, Basel, 2013, pp. 17–39.Google Scholar
Ball, J. A. and Bolotnikov, V., de Branges-Rovnyak spaces: basics and theory. Preprint, 2021. arXiv:1405.2980 Google Scholar
Chu, C., Density of polynomials in sub-Bergman Hilbert spaces. J. Math. Anal. Appl. 467(2018), no. 1, 699–703.CrossRefGoogle Scholar
Fricain, E. and Mashreghi, J., The theory of $\mathbf{\mathcal{H}}(b)$ spaces. Vol. 2. New Mathematical Monographs, 21, Cambridge University Press, Cambridge, 2016.Google Scholar
Garcia, S. R., Mashreghi, J., and Ross, W. T., Introduction to model spaces and their operators. Cambridge Studies in Advanced Mathematics, 148, Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar
Jury, M., Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators. Proc. Amer. Math. Soc. 135(2007), no. 11, 3669–3675.CrossRefGoogle Scholar
Micchelli, C. A., Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2(1986), no. 1, 11–22.CrossRefGoogle Scholar
Nikolski, N. K., Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz. Translated from the French by Andreas Hartmann, Mathematical Surveys and Monographs, 92, American Mathematical Society, Providence, RI, 2002.Google Scholar
Nowak, M. and Rososzczuk, R., Weighted sub-Bergman Hilbert spaces. Ann. Univ. Mariae Curie-Skłodowska Sect. A 68(2014), no. 1, 49–57.Google Scholar
Paulsen, V. I. and Raghupathi, M., An introduction to the theory of reproducing kernel Hilbert spaces. Cambridge Studies in Advanced Mathematics, 152, Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar
Rasmussen, C. E. and Williams, C. K. I., Gaussian processes for machine learning . In: Adaptive computation and machine learning, MIT Press, Cambridge, MA, 2006.Google Scholar
Sarason, D., Sub-Hardy Hilbert spaces in the unit disk. University of Arkansas Lecture Notes in the Mathematical Sciences, 10, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1994.Google Scholar
Sultanic, S., Sub-Bergman Hilbert spaces. J. Math. Anal. Appl. 324(2006), no. 1, 639–649.CrossRefGoogle Scholar
Zhu, K., Sub-Bergman Hilbert spaces on the unit disk. Indiana Univ. Math. J. 45(1996), no. 1, 165–176.CrossRefGoogle Scholar
Zhu, K., Sub-Bergman Hilbert spaces in the unit disk. II. J. Funct. Anal. 202(2003), no. 2, 327–341.Google Scholar