Abstract Interpolation in Vector-Valued de Branges-Rovnyak Spaces

作者：Ball Joseph A; Bolotnikov Vladimir^*; ter Horst Sanne

来源：Integral Equations and Operator Theory, 2011, 70(2): 227-263.

DOI：10.1007/s00020-010-1844-1

• de Branges-Rovnyak space
• abstract interpolation problem
• boundary interpolation
• operator-argument interpolation
• Redheffer transformations
• Toeplitz kernels

Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-96, Naukova Dumka, Kiev, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space H(K-s) associated with an operator-valued Schur class function S. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in H(K-S). In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.

• 出版日期2011-6
• 单位Virginia Tech; 美国弗吉尼亚理工大学(Virginia Tech)