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

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.