On a local Darlington synthesis problem

TL;DR

This paper introduces a local version of the Darlington synthesis problem, relating algebraic embedding conditions to analytic pseudocontinuation, and proves a local analog of the Arov--Douglas--Helton theorem.

Contribution

It formulates and proves a local version of the Darlington synthesis problem and extends the ADH theorem to a localized setting.

Findings

Established a local Darlington synthesis framework

Proved a local analog of the ADH theorem

Connected algebraic and analytic properties in a localized context

Abstract

The Darlington synthesis problem (in the scalar case) is the problem of embedding a given contractive analytic function to an inner $2\times 2$ matrix function as the entry. A fundamental result of Arov--Douglas--Helton relates this algebraic property to a pure analytic one known as a pseudocontinuation of bounded type. We suggest a local version of the Darlington synthesis problem and prove a local analog of the ADH theorem.

Equations

$$I-S^{\ast }(z)S(z)\ge 0,z\in \mathbb{D},$$

$$S(z)=\Vert s_{ij}(z){\Vert }_{i,j=1}^2:s_{22}(z)=s(z).$$

$$f(z)=\frac{f_1(z)}{f_2(z)},f_j\in \mathcal{S}(\Omega ).$$

$$f(t)=\tilde{f}(t)a.e.\mathrm{on}\gamma .$$

$$S(z)=\left[\begin{array}{cc}{s}_{11}(z)& 0\\ 0& s(z)\end{array}\right]$$

$${\int }_{\mathbb{T}}\log (1-|s(t){|}^2)m(dt)>-\infty ,$$

$$\tilde{a}(\zeta ):=\frac{1}{\overline{a(1/\bar{\zeta })}},\zeta \in {\mathbb{D}}_e.$$

$$\begin{array}{rl}U(\zeta )& :=(S^{-1}{)}^{\ast }(1/\bar{\zeta })\\ & =\frac{1}{\overline{\det S(1/\bar{\zeta })}}\left[\begin{array}{cc}\overline{s(1/\bar{\zeta })}& -\overline{s_{21}(1/\bar{\zeta })}\\ -\overline{s_{12}(1/\bar{\zeta })}& \overline{s_{11}(1/\bar{\zeta })}\end{array}\right],\zeta \in {\mathbb{D}}_e.\end{array}$$

$$\tilde{s}(\zeta )=\frac{\overline{s_{11}(1/\bar{\zeta })}}{\overline{\det S(1/\bar{\zeta })}}.$$

$$g(z):=\overline{\tilde{s}(1/\bar{z})}\in N(\mathbb{D}),h(z):=1-g(z)s(z)\in N(\mathbb{D}),$$

$${\int }_{\gamma }\log |h(t)|m(dt)={\int }_{\gamma }\log (1-|s(t){|}^2)m(dt)>-\infty .$$

$$\sigma_{\gamma}(z):=\exp\Bigl{\{}\frac{1}{2}\,\int_{\gamma}\frac{t+z}{t-z}\,\log(1-|s(t)|^{2})\,m(dt)\Bigr{\}}$$

$$|{\sigma }_{\gamma }(t){|}^2=1-|s(t){|}^{2}a.e.\mathrm{on}\gamma ,|{\sigma }_{\gamma }(t)|=1a.e.\mathrm{on}{\gamma }^{\prime }:=\mathbb{T}\backslash \gamma .$$

$$\begin{array}{rl}g(z)& =\frac{g_1(z)}{g_2(z)}=\frac{I_{g_1}(z)O_{g_1}(z)}{I_{g_2}(z)O_{g_2}(z)},g_j\in \mathcal{S},\\ h(z)& =\frac{h_1(z)}{h_2(z)}=\frac{I_{h_1}(z)O_{h_1}(z)}{I_{h_2}(z)O_{h_2}(z)},h_j\in \mathcal{S},\end{array}$$

$$\begin{split}O(z)&=O(z,\gamma)O(z,\gamma^{\prime}),\\ O(z,\Gamma)&:=\exp\Bigl{\{}\frac{1}{2}\,\int_{\Gamma}\frac{t+z}{t-z}\,\log|O(t)|\,m(dt)\Bigr{\}}\end{split}$$

$$|O(t,\gamma )|=1a.e.\mathrm{on}{\gamma }^{\prime },|O(t,{\gamma }^{\prime })|=1a.e.\mathrm{on}\gamma .$$

$$\begin{array}{rl}g(z)& =\frac{I_{g_1}(z)O_{g_1}(z,\gamma )O_{g_1}(z,{\gamma }^{\prime })}{I_{g_2}(z)O_{g_2}(z,\gamma )O_{g_2}(z,{\gamma }^{\prime })},\\ h(z)& =\frac{I_{h_1}(z)O_{h_1}(z,\gamma )O_{h_1}(z,{\gamma }^{\prime })}{I_{h_2}(z)O_{h_2}(z,\gamma )O_{h_2}(z,{\gamma }^{\prime })}.\end{array}$$

$$p(z):=I_{g_2}(z)I_{h_2}(z)O_{g_2}(z,{\gamma }^{\prime })O_{h_2}(z,{\gamma }^{\prime }),$$

$$\begin{array}{rl}{s}_{11}(z)& =-p(z)g(z)=-I_{g_1}(z)I_{h_2}(z)O_{g_1}(z,{\gamma }^{\prime })O_{h_2}(z,{\gamma }^{\prime })\frac{O_{g_1}(z,\gamma )}{O_{g_2}(z,\gamma )},\\ s_{21}(z)& =p(z)\frac{h(z)}{{\sigma }_{\gamma }(z)}=I_{g_2}(z)I_{h_1}(z)O_{g_2}(z,{\gamma }^{\prime })O_{h_1}(z,{\gamma }^{\prime })\frac{O_{h_1}(z,\gamma )}{O_{h_2}(z,\gamma ){\sigma }_{\gamma }(z)}.\end{array}$$

$$f\in N^{+}(\mathbb{D}),|f(t)|\le 1a.e.\mathrm{on}\mathbb{T}\Rightarrow f\in \mathcal{S}.$$

$$|s_{11}(t)|\le |g(t)|=|s(t)|\le 1,|s_{21}(t)|\le \frac{|h(t)|}{|{\sigma }_{\gamma }(t)|}=(1-|s(t){|}^2{)}^{1/2}\le 1,$$

$$S^{\ast }(t)S(t)=\left[\begin{array}{cc}|s_{11}(t){|}^2+|s_{21}(t){|}^2& \overline{s_{11}(t)}{s}_{12}(t)+\overline{s_{21}(t)}s(t)\\ s_{11}(t)\overline{s_{12}(t)}+s_{21}(t)\overline{s(t)}& |s_{12}(t){|}^2+|s(t){|}^2\end{array}\right].$$

$$|s_{12}(t){|}^2+|s(t){|}^2=|{\sigma }_{\gamma }(t){|}^2+|s(t){|}^2=1-|s(t){|}^2+|s(t){|}^2=1$$

$$|s_{11}(t){|}^2+|s_{21}(t){|}^2=|g(t){|}^2+\frac{|h(t){|}^2|}{|{\sigma }_{\gamma }(t){|}^2}=|s(t){|}^2+1-|s(t){|}^2=1.$$

$$\begin{split}s_{11}(t)\overline{s_{12}(t)}+s_{21}(t)\overline{s(t)}&=p(t)\Bigl{(}-g(t)\overline{\sigma_{\gamma}(t)}+\frac{h(t)}{\sigma_{\gamma}(t)}\,\overline{s(t)}\Bigr{)}\\ &=p(t)\overline{s(t)}\Bigl{(}-\overline{\sigma_{\gamma}(t)}+\frac{h(t)}{\sigma_{\gamma}(t)}\Bigr{)}=0\end{split}$$

$$I-V^{\ast }(t)V(t)=\left[\begin{array}{cc}\ast & \ast \\ \ast & 1-|v_{12}(t){|}^2-|s(t){|}^2\end{array}\right]\ge 0,t\in \mathbb{T},$$

$$\sigma_{\gamma^{\prime}}(z):=\exp\Bigl{\{}\frac{1}{2}\,\int_{\gamma^{\prime}}\frac{t+z}{t-z}\,\log(1-|s(t)|^{2})\,m(dt)\Bigr{\}}$$

$$|e(t)|=1a.e.\mathrm{on}\gamma ,|e(t)|=\epsilon a.e.\mathrm{on}{\gamma }^{\prime },$$

$$\begin{array}{rl}V(z)& =\left[\begin{array}{cc}r(z)& 0\\ 0& 1\end{array}\right]S(z)\left[\begin{array}{cc}r(z)& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}{r}^2(z)s_{11}(z)& r(z)s_{12}(z)\\ r(z)s_{21}(z)& s(z)\end{array}\right],\end{array}$$

$$W=\left[\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right]:=I-V^{\ast }V=\left[\begin{array}{cc}1-|r^2{s}_{11}{|}^2-|r{s}_{21}{|}^2& -\bar{r}|r{|}^2{s}_{12}\overline{s_{11}}-\overline{r{s}_{21}}s\\ -r|r{|}^2{s}_{11}\overline{s_{12}}-r{s}_{21}\overline{s}& 1-|r{s}_{12}{|}^2-|s{|}^2\end{array}\right].$$

Full text

Introduction

The Darlington synthesis with its origin in electrical engineering has a long history. The synthesis of non-lossless circuits was a hard problem at the pre-computer time. The idea of the Darlington synthesis was to reduce any such problem to a lossless one

A mathematical setup in the simplest scalar case looks as follows, see [1, 2, 3, 5] and [6, Section 6.7].

An analytic function $s$ on the unit disk ${\mathbb{D}}$ is called a Schur (contractive) function, $s\in{\mathcal{S}}$, if $|s|\leq 1$ in ${\mathbb{D}}$. Similarly, an analytic on ${\mathbb{D}}$ $2\times 2$ matrix function $S$ (throughout this note we deal only with matrices of order $2$) is a Schur (contractive) matrix function, $S\in{\mathcal{S}}^{(m)}$, if

[TABLE]

$I$ is a unity matrix. A function $s\in{\mathcal{S}}$ (a matrix function $S\in{\mathcal{S}}^{(m)}$) is said to be inner (matrix) function if its boundary values which exist almost everywhere on the unit circle ${\mathbb{T}}$, are unimodular (unitary). Given $s\in{\mathcal{S}}$, the Darlington synthesis problem asks whether there exists an inner matrix function $S\in{\mathcal{S}}^{(m)}$ so that

[TABLE]

A seminal result of Arov [1] and Douglas–Helton [3] states that a Schur function admits the Darlington synthesis if and only if it possesses a pseudocontinuation of bounded type across ${\mathbb{T}}$. Recall that a meromorphic on a region $\Omega$ function of bounded type is the quotient of two bounded (or contractive) analytic on $\Omega$ functions

[TABLE]

Such functions constitute the Nevanlinna class $N(\Omega)$.

The goal of this note is to suggest a local version of the Darlington synthesis problem and to prove a local analog of the Arov–Douglas–Helton theorem.

Definition 0.1**.**

Let $\gamma$ be an arc of the unit circle (the case $\gamma={\mathbb{T}}$ is not excluded). Denote by ${\mathbb{D}}_{e}$ the exterior of the unit disk with respect to the extended complex plane $\bar{\mathbb{C}}$. A function $f\in N({\mathbb{D}})$ admits the pseudocontinuation of bounded type across $\gamma$ if there is a function $\tilde{f}\in N({\mathbb{D}}_{e})$ so that their boundary values agree

[TABLE]

We write $f\in PC_{\gamma}$ for such functions. The class $PC_{\gamma}$ is nontrivial, see Example 1.1 in Section 1.

Theorem 0.2**.**

Let $s\in{\mathcal{S}}$. The following conditions are equivalent.

1. (1)

There is a matrix function $S=\|s_{ij}\|_{i,j=1}^{2}$ so that $s_{ij}\in{\mathcal{S}}$, $s_{22}=s$, and $S$ is unitary a.e. on the arc $\gamma$; 2. (2)

$s\in PC_{\gamma}$.

In the case $\gamma={\mathbb{T}}$, the above matrix function $S$ is inner due to the Maximum Norm Principle, and we come to the Arov–Douglas–Helton theorem.

Given an arc $\gamma$, we denote by ${\mathcal{S}}_{\gamma}$ ($N_{\gamma}$) the class of the Schur (Nevanlinna) functions, unimodular a.e. on $\gamma$. Similarly, ${\mathcal{S}}^{(m)}_{\gamma}$ stands for the class of the Schur matrix functions unitary a.e. on $\gamma$.

It is clear that a matrix function with contractive entries does not necessarily belong to ${\mathcal{S}}^{(m)}$. So the question arises naturally whether the matrix $S$ in Theorem 0.2 can be taken from ${\mathcal{S}}^{(m)}_{\gamma}$. If $s\in{\mathcal{S}}_{\gamma}$, the answer is affirmative: the matrix function

[TABLE]

with an arbitrary inner function $s_{11}$ belongs to ${\mathcal{S}}^{(m)}_{\gamma}$. But, in general, the answer is negative. The reason is that $s$ being an entry of a contractive, nondiagonal matrix function is supposed to obey a global condition

[TABLE]

$m$ is the normalized Lebesgue measure on ${\mathbb{T}}$. As it turns out, this condition is also sufficient.

Theorem 0.3**.**

Let $s\in{\mathcal{S}}^{\prime}_{\gamma}={\mathcal{S}}\backslash{\mathcal{S}}_{\gamma}$. The following conditions are equivalent.

1. (1)

There is a matrix function $V=\|v_{ij}\|_{i,j=1}^{2}\in{\mathcal{S}}^{(m)}_{\gamma}$ so that $v_{22}=s$; 2. (2)

$s\in PC_{\gamma}$* and $\eqref{logint}$ holds.*

In contrast to the case $\gamma={\mathbb{T}}$ of the whole unit circle, we have neither the model spaces theory nor the Douglas–Shapiro–Shilds theorem at hand. So the argument is more or less straightforward and relies upon the explicit (in a sense) expressions for the matrix entries of the matrices in question.

1. Local pseudocontinuation and Darlington synthesis

Let us begin with the classes ${\mathcal{S}}_{\gamma}$ and $N_{\gamma}$, which play the same role as the class of inner functions does in the classical setting of the Darlington synthesis problem.

Example 1.1**.**

Let $a\in N_{\gamma}$. Write

[TABLE]

Then $\tilde{a}\in N({\mathbb{D}}_{e})$ and $\tilde{a}=a$ a.e. on $\gamma$, so $N_{\gamma}\subset PC_{\gamma}$. In particular, ${\mathcal{S}}_{\gamma}\subset PC_{\gamma}$, and, moreover, such $s\in PC_{\gamma}\backslash PC_{\mathbb{T}}$ unless $s$ is an inner function.

Proof of Theorem 0.2.

$(1)\Rightarrow(2)$. The argument here is standard. By the hypothesis, $\det S\not\equiv 0$, so we write

[TABLE]

It is clear that all entries of $U$ belong to $N({\mathbb{D}}_{e})$, and $U=S$ a.e. on $\gamma$. Hence, $s$ admits the pseudocontinuation of bounded type across $\gamma$, $s\in PC_{\gamma}$, with

[TABLE]

Note that in fact each entry of the bounded matrix function $S$, unitary a.e. on $\gamma$, is in the class $PC_{\gamma}$.

$(2)\Rightarrow(1)$. The arguments in Example 1.1 and around relation (0.4) show that the result holds for $s\in{\mathcal{S}}_{\gamma}$. So we assume further that $s\in{\mathcal{S}}^{\prime}_{\gamma}={\mathcal{S}}\backslash{\mathcal{S}}_{\gamma}$.

Define a pair of functions on ${\mathbb{D}}$

[TABLE]

where $\tilde{s}$ is the pseudocontinuation of bounded type of $s$ across $\gamma$. Now, $s\notin{\mathcal{S}}_{\gamma}$ implies $h\not\equiv 0$, so $\log|h|\in L^{1}({\mathbb{T}})$, see [4, Theorem 2.2], and

[TABLE]

We see that $\log(1-|s|^{2})\in L^{1}(\gamma)$ as long as $s\in PC_{\gamma}\backslash{\mathcal{S}}_{\gamma}$, which is a local counterpart of relation (0.5).

In view of (1.4), the function

[TABLE]

is a well-defined, outer Schur function, $\sigma_{\gamma}\in{\mathcal{S}}_{out}$, with the boundary values

[TABLE]

We choose $s_{12}=\sigma_{\gamma}$.

Going back to the Nevanlinna functions $g,h$ in (1.3), we write

[TABLE]

where $f=I_{f}O_{f}$ is the standard inner-outer factorization of a Schur function $f$. We proceed with the further factorization of the outer factors with respect to $\gamma$, precisely,

[TABLE]

for the arc $\Gamma=\gamma,\gamma^{\prime}$. We have $O(\cdot,\Gamma)\in{\mathcal{S}}_{out}$ and

[TABLE]

Hence,

[TABLE]

Put

[TABLE]

so $|p|=1$ a.e. on $\gamma$. Our choice of $s_{11}$ and $s_{21}$ is

[TABLE]

It is clear that $s_{22}=s$ and $s_{12}=\sigma_{\gamma}$ (1.5) are contractive functions. As for $s_{11}$ and $s_{21}$ (1.11), we note that they belong to an important subclass $N^{+}({\mathbb{D}})\subset N({\mathbb{D}})$ of the Nevanlinna class, which is usually referred to as the Smirnov class, see [4, Section 2.5]. It is characterized by the denominator in (0.2) being an outer Schur function, which is exactly the case in (1.11). The main feature of this class is the Smirnov maximum modulus principle, [4, Theorem 2.11],

[TABLE]

For $t\in\gamma^{\prime}$ we have $|s_{11}|\leq 1$, $|s_{21}|\leq 1$ a.e. in view of (1.8). For $t\in\gamma$ we have $|p|=1$ a.e., so

[TABLE]

and the first claim of the Theorem follows from (1.12).

To show that $S$ is unitary a.e. on $\gamma$, we put

[TABLE]

By (1.6),

[TABLE]

a.e. on $\gamma$. Next, $|p|=1$ a.e. on $\gamma$ implies

[TABLE]

Finally, by (1.6) and the definition of $h$,

[TABLE]

a.e. on $\gamma$. So, $S^{*}S=I$, as claimed. The proof is complete. $\Box$

Proof of Theorem 0.3.

$(1)\Rightarrow(2)$. By Theorem 0.2, $s\in PC_{\gamma}$, so we have to verify condition (0.5). Note that at least one of the functions $v_{12}$, $v_{21}$ is not identically zero (otherwise, $s\in{\mathcal{S}}_{\gamma}$). Assume that $v_{12}\not\equiv 0$ and write

[TABLE]

so $1-|s(t)|^{2}\geq|v_{12}(t)|^{2}$. Since $\log|v_{12}|\in L^{1}({\mathbb{T}})$, condition (0.5) follows.

$(2)\Rightarrow(1)$. The matrix $V$ arises as an appropriate modification of the matrix $S$ from Theorem 0.2. By (0.5), the function

[TABLE]

is well-defined and lies in ${\mathcal{S}}_{out}$. Denote by $e$ the outer Schur function with

[TABLE]

where $0<\varepsilon<1/3$ is a small enough positive constant, and put $r:=e\sigma_{\gamma^{\prime}}$. Take the matrix $V$ in question as

[TABLE]

As both $e$ and $\sigma_{\gamma^{\prime}}$ are unimodular on $\gamma$, then so is $r$, and thereby $V$ is unitary a.e. on $\gamma$.

It remains to check that $V\in{\mathcal{S}}^{(m)}$. To this end we put on the arc $\gamma^{\prime}$

[TABLE]

Since

[TABLE]

then $|rs_{12}|^{2}=\varepsilon^{2}(1-|s|^{2})$ and so

[TABLE]

a.e. on $\gamma^{\prime}$. Next, the functions $r,s_{11},s_{21}$ are contractive, so

[TABLE]

and

[TABLE]

a.e. on $\gamma^{\prime}$ for $0<\varepsilon<1/3$. Finally,

[TABLE]

and so

[TABLE]

a.e. on $\gamma^{\prime}$.

To show that $\widetilde{W}\geq 0$ a.e. on $\gamma^{\prime}$, given $\tilde{w}_{11}\geq 0$, $\tilde{w}_{22}\geq 0$, we compute the determinant of $\widetilde{W}$

[TABLE]

a.e. on $\gamma^{\prime}$ for $0<\varepsilon<1/3$. So, $V\in{\mathcal{S}}^{(m)}_{\gamma}$, as claimed. $\Box$

We complete this note with some properties of the pseudocontinuation of bounded type across an arc.

Proposition 1.2**.**

Let $s_{1},s_{2}\in{\mathcal{S}}$ and $|s_{1}|=|s_{2}|$ a.e. on the arc $\gamma$. Then $s_{1}$ and $s_{2}$ belong to $PC_{\gamma}$ simultaneously.

Proof.

Let $s_{1}\in PC_{\gamma}$. We have the canonical factorization

[TABLE]

and, by the assumption, $O_{1}(\cdot,\gamma)=O_{2}(\cdot,\gamma)$. Hence,

[TABLE]

The function $a\in N_{\gamma}$, so, see Example 1.1, $a\in PC_{\gamma}$. The later class is closed under multiplication, so $s_{2}\in PC_{\gamma}$, as claimed. ∎

Recall that $\sigma_{\gamma}$ is defined in (1.5) under condition (1.4).

Proposition 1.3**.**

Let $s\in{\mathcal{S}}$ and $\log(1-|s|^{2})\in L^{1}(\gamma)$. Then

[TABLE]

Proof.

As we mentioned earlier in the proof of Theorem 0.2, each entry of the bounded matrix function $S$, unitary a.e. on $\gamma$, is in the class $PC_{\gamma}$. If $s\in PC_{\gamma}$, the matrix function $S$ in Theorem 0.2 contains both $s$ and $\sigma_{\gamma}$ as its entries, and we are done.

Conversely, let $\sigma_{\gamma}\in PC_{\gamma}$. By Theorem 0.2, there is a matrix function $\Sigma$ with contractive entries, unitary a.e. on $\gamma$, and

[TABLE]

In particular, $|\sigma_{12}|^{2}+|\sigma_{\gamma}|^{2}=1$, and so $|\sigma_{12}|=|s|$ a.e. on $\gamma$. The function $\sigma_{12}$, being the entry of $\Sigma$, belongs to the class $PC_{\gamma}$. By Proposition 1.2, so does $s$, as claimed. ∎

Remark 1.4**.**

The fact that $\gamma$ is the arc of the unit circle is obviously immaterial. The argument works for an arbitrary Borel set $\gamma\subset{\mathbb{T}}$ of positive measure.

Acknowledgement. The author thanks the participants of the Analysis Seminar at Kharkov National University for valuable discussions.