Remarks on the Operator-valued Interpolation for Multivariable Bounded Analytic Functions

We consider a domain D ⊂ ℂn defined by a uniform norm inequality supλ ∥Δλ(z)∥ < 1 involving a set Δ = (Δλ)λ of matrix-valued analytic functions Δλ. The associated Schur interpolation class is then $S=\left \{ \right.F=F(z):\sup_{||\Delta (Z)||\textless1}||F(Z)||\leq 1\left. \right \}$, where Z ru...

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Bibliographic Details
Published in:Indiana University mathematics journal 2004-01, Vol.53 (6), p.1551-1576
Main Author: Ambrozie, C.-G.
Format: Article
Language:English
Subjects:
Analytic functions
Commuting
Exact sciences and technology
Hilbert spaces
Interpolation
Linear transformations
Mathematical analysis
Mathematical functions
Mathematical inequalities
Mathematical theorems
Mathematics
Operator theory
Sciences and techniques of general use
Unit ball
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Summary:We consider a domain D ⊂ ℂn defined by a uniform norm inequality supλ ∥Δλ(z)∥ < 1 involving a set Δ = (Δλ)λ of matrix-valued analytic functions Δλ. The associated Schur interpolation class is then $S=\left \{ \right.F=F(z):\sup_{||\Delta (Z)||\textless1}||F(Z)||\leq 1\left. \right \}$, where Z runs the commuting n-tuples of matrices. We characterize by positive-definiteness conditions the existence of the solutions F ∈ S of an operator-valued Nevanlinna-Pick type problem over D. Also, we describe the elements of S as fractional transforms F = a22 + a21(I − Δa11)−1 Δa12, with $[a_{ij}]^2_{i,j=1}$ unitary. The results are based on a representation technique due to J. Agler [1].
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2004.53.2472