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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

$ J_{\lowercase{p},\lowercase{m}}$-inner dilations of matrix-valued functions that belong to the Carathéodory class and admit pseudocontinuation


Authors: D. Z. Arov and N. A. Rozhenko
Translated by: V. Vasyunin
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 375-395
MSC (2000): Primary 47A56
Published electronically: March 21, 2008
MathSciNet review: 2340706
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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}:\vert z\vert<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<\vert z\vert\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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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-08-01002-9
PII: S 1061-0022(08)01002-9
Keywords: Holomorphic matrix-valued functions, dilations, pseudocontinuation
Received by editor(s): November 9, 2006
Published electronically: March 21, 2008
Article copyright: © Copyright 2008 American Mathematical Society