de Branges–Rovnyak Spaces: Basics and Theory

Joseph A. Ball () and Vladimir Bolotnikov ()
Additional contact information
Joseph A. Ball: Virginia Tech, Department of Mathematics
Vladimir Bolotnikov: The College of William and Mary, Department of Mathematics

Chapter 27 in Operator Theory, 2015, pp 631-679 from Springer

Abstract: Abstract For S a contractive analytic operator-valued function on the unit disk 𝔻 $$\mathbb{D}$$ , de Branges and Rovnyak associate a Hilbert space of analytic functions ℋ ( S ) $$\mathcal{H}(S)$$ and related extension space 𝒟 ( S ) $$\mathcal{D}(S)$$ consisting of pairs of analytic functions on the unit disk 𝔻 $$\mathbb{D}$$ . This survey describes three equivalent formulations (the original geometric de Branges–Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges–Rovnyak space ℋ ( S ) $$\mathcal{H}(S)$$ , de Branges–Rovnyak spaces as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges–Rovnyak model space 𝒟 ( S ) $$\mathcal{D}(S)$$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article.

Keywords: Reproducing Kernel Hilbert Space (RKHS); Toeplitz Operators; Contraction Operator; Pullback Space; Schur-class Function (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0667-1_6

Ordering information: This item can be ordered from
http://www.springer.com/9783034806671

DOI: 10.1007/978-3-0348-0667-1_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().