Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting. (English) Zbl 1303.47016
Having in mind the classical case concerning the scalar-valued functions, and given two coefficient Hilbert space \(\mathcal{U}_T\) and \(\mathcal{Y}_T\), a family \(\Psi\) of functions \(\psi\) on a set \(\Omega\) with values in \(\mathcal{L}(\mathcal{U}_T,\mathcal{Y}_T)\) is said to be a collection of test functions if \(\sup \{\|\psi(z)\|:\psi\in\Psi\}<1\) for each \(z\in\Omega\). The associated Schur-Agler class is the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. The authors present extensions of the classical framework to this generalized framework, that is, the case where the test functions, kernel functions, and Schur-Agler class are allowed to be matrix-valued or operator-valued. The general theory obtained in this way is illustrated by some examples, emphasizing the case when the matrix-valued version is not obtained by tensoring the scalar-valued one.
MSC:
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 47A57 | Linear operator methods in interpolation, moment and extension problems |
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
Schur-Agler class; test functions; positive kernels; completely positive kernels; reproducing kernel Hilbert spaces; transfer-function realization; internal tensor product of correspondences; unitary colligation matrix; separation of convex sets; interior point of convex hull; finitely connected planar domain; constrained \(H^\infty\)-algebraReferences:
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