$J_{p,m}$-inner dilations of matrix-valued functions that belong to the Carathéodory class and admit pseudocontinuation
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D. Z. Arov and N. A. Rozhenko
Translated by: V. Vasyunin - St. Petersburg Math. J. 19 (2008), 375-395
- DOI: https://doi.org/10.1090/S1061-0022-08-01002-9
- Published electronically: March 21, 2008
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Abstract:
The class $\ell ^{p\times p}$ of matrix-valued functions $c(z)$ holomorphic in the unit disk $D=\{{z\in \mathbb {C}:|z|<1}\}$, having order $p$, and satisfying $\operatorname {Re}c(z)\ge 0$ in $D$ is considered, as well as its subclass $\ell ^{p\times p}\Pi$ of matrix-valued functions $c(z)\in \ell ^{p\times p}$ that have a meromorphic pseudocontinuation $c_-(z)$ to the complement $D_e=\{z\in \mathbb {C}:1<|z|\le \infty \}$ of the unit disk with bounded Nevanlinna characteristic in $D_e$.
For matrix-valued functions $c(z)$ of class $\ell ^{p\times p}\Pi$ a representation as a block of a certain $J_{p,m}$-inner matrix-valued function $\theta (z)$ is obtained. The latter function has a special structure and is called the $J_{p,m}$-inner dilation of $c(z)$. The description of all such representations is given.
In addition, the following special $J_{p,m}$-inner dilations are considered and described: minimal, optimal, $*$-optimal, minimal and optimal, minimal and $*$-optimal. Also, $J_{p,m}$-inner dilations with additional properties are treated: real, symmetric, rational, or any combination of them under the corresponding restrictions on the matrix-valued function $c(z)$. The results extend to the case where the open upper half-plane $\mathbb {C}_+$ is considered instead of the unit disk $D$. For entire matrix-valued functions $c(z)$ with $\operatorname {Re}c(z) \ge 0$ in $\mathbb {C_+}$ and with Nevanlinna characteristic in $\mathbb {C}_-$, the $J_{p,m}$-inner dilations in $\mathbb {C}_+$ that are entire matrix-valued functions are also described.
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Bibliographic Information
- D. Z. Arov
- Affiliation: Department of Physics and Mathematics, South-Ukrainian State Pedagogical University, Staroportofrankovskaya, 26, 650029, Odessa, Ukraine
- N. A. Rozhenko
- Affiliation: Department of Physics and Mathematics, South-Ukrainian State Pedagogical University, Staroportofrankovskaya, 26, 650029, Odessa, Ukraine
- Received by editor(s): November 9, 2006
- Published electronically: March 21, 2008
- © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 375-395
- MSC (2000): Primary 47A56
- DOI: https://doi.org/10.1090/S1061-0022-08-01002-9
- MathSciNet review: 2340706