# Hilbert Spaces Contractively Included in the Hardy Space of the Bidisk

Hilbert Spaces Contractively Included in the Hardy Space of the Bidisk We study the reproducing kernel Hilbert spaces $$\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)$$ with kernels of the form $$\frac{{I - S(z_1 ,z_2 > )S(w_1 ,w_2 )^* }}{{(1 - z_1 w_1^* )(1 - z_2 w_2^* )}}$$ where S(z1,z2) is a Schur function of two variables z 1,z2ℓ $$\mathbb{D}$$ . They are analogs of the spaces $$\mathfrak{H}\left( {\mathbb{D},S} \right)$$ with reproducing kernel (1-S(z)S(w)*)/(1-zw*) introduced by de Branges and Rovnyak l. de Branges and J. Rovnyak, Square Summable Power Series Holt, Rinehart and Winston, New York, 1966. We discuss the characterization of $$\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)$$ as a subspace of the Hardy space on the bidisk. The spaces $$\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)$$ form a proper subset of the class of the so–called sub–Hardy Hilbert spaces of the bidisk. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Hilbert Spaces Contractively Included in the Hardy Space of the Bidisk

, Volume 5 (1) – Oct 3, 2004
26 pages

/lp/springer_journal/hilbert-spaces-contractively-included-in-the-hardy-space-of-the-bidisk-00VJ8OeRql
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009826406222
Publisher site
See Article on Publisher Site

### Abstract

We study the reproducing kernel Hilbert spaces $$\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)$$ with kernels of the form $$\frac{{I - S(z_1 ,z_2 > )S(w_1 ,w_2 )^* }}{{(1 - z_1 w_1^* )(1 - z_2 w_2^* )}}$$ where S(z1,z2) is a Schur function of two variables z 1,z2ℓ $$\mathbb{D}$$ . They are analogs of the spaces $$\mathfrak{H}\left( {\mathbb{D},S} \right)$$ with reproducing kernel (1-S(z)S(w)*)/(1-zw*) introduced by de Branges and Rovnyak l. de Branges and J. Rovnyak, Square Summable Power Series Holt, Rinehart and Winston, New York, 1966. We discuss the characterization of $$\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)$$ as a subspace of the Hardy space on the bidisk. The spaces $$\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)$$ form a proper subset of the class of the so–called sub–Hardy Hilbert spaces of the bidisk.

### Journal

PositivitySpringer Journals

Published: Oct 3, 2004

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