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.
Citation:
D. Z. Arov, N. A. Rozhenko, “$J_{p,m}$-inner dilations of matrix-valued functions that belong to the Carathódory class and admit pseudocontinuation”, Algebra i Analiz, 19:3 (2007), 76–105; St. Petersburg Math. J., 19:3 (2008), 375–395
\Bibitem{AroRoz07}
\by D.~Z.~Arov, N.~A.~Rozhenko
\paper $J_{p,m}$-inner dilations of matrix-valued functions that belong to the Carath\'odory class and admit pseudocontinuation
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 3
\pages 76--105
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\zmath{https://zbmath.org/?q=an:1210.47038}
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\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 3
\pages 375--395
\crossref{https://doi.org/10.1090/S1061-0022-08-01002-9}
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Linking options:
https://www.mathnet.ru/eng/aa120
https://www.mathnet.ru/eng/aa/v19/i3/p76
This publication is cited in the following 4 articles:
Fei J., Yeh Ch.-N., Zgid D., Gull E., “Analytical Continuation of Matrix-Valued Functions: Caratheodory Formalism”, Phys. Rev. B, 104:16 (2021), 165111
Didenko V.D., Rozhenko N.A., “A Class of Stationary Stochastic Processes”, Studia Math., 222:3 (2014), 191–205
D. Z. Arov, N. A. Rozhenko, “On the Relation between the Darlington Realizations of Matrix Functions from the Carathéodory Class and Their $J_{p,r}$-Inner SI-Dilations”, Math. Notes, 90:6 (2011), 801–812
D. Z. Arov, N. A. Rozhenko, “To the theory of passive systems of resistance with losses of scattering channels”, J. Math. Sci. (N. Y.), 156:5 (2009), 742–760