The Schur Problem and Block Operator CMV Matrices

Abstract

The block operator CMV matrices and their sub-matrices are applied to the description of all solutions to the Schur interpolation problem for contractive analytic operator-valued functions in the unit disk (the Schur class functions).

Appendix A: Special Cases of Block Operator CMV Matrices

Let $\{\Gamma _n\}$ be the Schur parameters of the function $\Theta \in \mathbf{S}(\mathfrak{M },\mathfrak{N })$. Suppose $\Gamma _m$ is an isometry (respect., co-isometry, unitary) for some $m\ge 0$. Then $\Theta _m(z)=\Gamma _m$ for all $z\in \mathbb{D }$ and

$$\begin{aligned}&\Theta _{m-1}(z)=\Gamma _{m-1}+z D_{\Gamma ^*_{m-1}}\Gamma _m(I_{\mathfrak{D }_{\Gamma _{m-1}}} +z\Gamma ^*_{m-1}\Gamma _m)^{-1}D_{\Gamma _{m-1}},\\&\Theta _{m-2}(z)=\Gamma _{m-2}+z D_{\Gamma ^*_{m-2}}\Theta _{m-1}(z)(I_{\mathfrak{D }_{\Gamma _{m-2}}}+ z\Gamma ^*_{m-2}\Theta _{m-1}(z))^{-1}D_{\Gamma _{m-2}},\\&\ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots ,\\&\Theta (z)=\Gamma _{0}+z D_{\Gamma ^*_{0}}\Theta _{1}(z)(I_{\mathfrak{D }_{\Gamma _0}}+ z\Gamma ^*_{0}\Theta _{1}(z))^{-1}D_{\Gamma _{0}},\; z\in \mathbb{D }. \end{aligned}$$

The function $\Theta$ is the transfer function of the simple conservative systems constructed by means of its Schur parameters $\{\Gamma _n\}$ and the corresponding block operator CMV matrices $\mathcal{U }_0$ and $\widetilde{\mathcal{U }}_0$ [6]. Here we present the explicit form of block operator CMV and truncated CMV matrices. In particular we revise some misprints in [6]. Notice that if $\Gamma _m$ is isometric (respect., co-isometric), then

• 1. $\mathfrak{D }_{\Gamma ^*_n}=\mathfrak{D }_{\Gamma ^*_{m}}, D_{\Gamma ^*_{n}}=I_{\mathfrak{D }_{\Gamma ^*_{m}}}, \Gamma _n=0:\{0\}\rightarrow \mathfrak{D }_{\Gamma ^*_{m}}$ for $n> m$ (respect., $\mathfrak{D }_{\Gamma _n}=\mathfrak{D }_{\Gamma _{m}}, D_{\Gamma _{n}}=I_{\mathfrak{D }_{\Gamma _{m}}}, \Gamma _n=0:\mathfrak{D }_{\Gamma _n}\rightarrow \{0\}$ for $n> m$);

• 2. in the definitions of the state spaces $\mathfrak{H }_0=\mathfrak{H }_0(\{\Gamma _n\}_{n\ge 0})$ and $\widetilde{\mathfrak{H }}_0=\widetilde{\mathfrak{H }}_0(\{\Gamma _n\}_{n\ge 0})$ we replace $\mathfrak{D }_{\Gamma _n}$ with $\{0\}$ (respect., $\mathfrak{D }_{\Gamma ^*_n}$ with $\{0\}$) for $n\ge m$, and $\mathfrak{D }_{\Gamma ^*_n}$ by $\mathfrak{D }_{\Gamma ^*_m}$ (respect., $\mathfrak{D }_{\Gamma _n}$ by $\mathfrak{D }_{\Gamma _m}$) for $n> m$.

• 3. the corresponding unitary elementary rotation takes the row (respect., the column) form, i.e,

  $$\begin{aligned} \mathbf{J}^{(r)}_{\Gamma _0}&= \begin{bmatrix} \Gamma _0&I_{\mathfrak{D }_{\Gamma ^*_0}} \end{bmatrix}: \begin{array}{l} \mathfrak{M }\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{0}} \end{array}\rightarrow \mathfrak{N }\\&\times \left( {\mathrm{respect}},\;\mathbf{J}^{(c)}_{\Gamma _0}=\begin{bmatrix} \Gamma _0\\D_{\Gamma _0} \end{bmatrix}: \mathfrak{M }\rightarrow \begin{array}{l} \mathfrak{N }\\ \oplus \\ \mathfrak{D }_{\Gamma _{0}} \end{array}\right) ,\\ \mathbf{J}^{(r)}_{\Gamma _m}&= \begin{bmatrix} \Gamma _m&I_{\mathfrak{D }_{\Gamma ^*_m}} \end{bmatrix}: \begin{array}{l} \mathfrak{D }_{\Gamma _{m-1}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{m}}\end{array} \rightarrow \mathfrak{D }_{\Gamma ^*_{m-1}}\\&\times \left( {\mathrm{respect.}},\; \mathbf{J}^{(c)}_{\Gamma _m}= \begin{bmatrix} \Gamma _m\\D_{\Gamma _m} \end{bmatrix}: \mathfrak{D }_{\Gamma _{m-1}}\rightarrow \begin{array}{l} \mathfrak{D }_{\Gamma ^*_{m-1}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{m}} \end{array}\right) ,\; m\ge 1. \end{aligned}$$

Therefore, in definitions (3.1) of the block diagonal operator matrices

$$\begin{aligned} \mathcal{L }_0=\mathcal{L }_0(\{\Gamma _n\}_{n\ge 0}),\; \mathcal{M }_0=\mathcal{M }_0(\{\Gamma _n\}_{n\ge 0}),\;{\mathrm{and}}\; \widetilde{\mathcal{M }}_0=\widetilde{\mathcal{M }}_0(\{\Gamma _n\}_{n\ge 0}) \end{aligned}$$

we will replace

• $\mathbf{J}_{\Gamma _m}$ by $\mathbf{J}^{(r)}_{\Gamma _m}$ and $\mathbf{J}_{\Gamma _n}$ by $I_{\mathfrak{D }_{\Gamma ^*_m}}$ for $n>m$, when $\Gamma _m$ is isometry,

• $\mathbf{J}_{\Gamma _m}$ by $\mathbf{J}^{(c)}_{\Gamma _m}$, and $\mathbf{J}_{\Gamma _n}$ by $I_{\mathfrak{D }_{\Gamma _m}}$ for $n>m$, when $\Gamma _m$ is co-isometry,

• $\mathbf{J}_{\Gamma _m}$ by $\Gamma _m$, when $\Gamma _m$ is unitary.

In all these cases the block operators CMV matrices $\mathcal{U }_0=\mathcal{U }_0(\{\Gamma _n\}_{n\ge 0})$ and $\widetilde{\mathcal{U }}_0=\widetilde{\mathcal{U }}_0(\{\Gamma _n\}_{n\ge 0})$ are defined by means the products $\mathcal{U }_0=\mathcal{L }_0\mathcal{M }_0,\;\widetilde{\mathcal{U }}_0=\widetilde{\mathcal{M }}_0\mathcal{L }_0.$ These matrices are five block-diagonal. In the case when the operator $\Gamma _m$ is unitary the block operator CMV matrices $\mathcal{U }_0$ and $\widetilde{\mathcal{U }}_0$ are finite and otherwise they are semi-infinite. As before the truncated block operator CMV matrices $\mathcal{T }_0=\mathcal{T }_0((\{\Gamma _n\}_{n\ge 0})$ and $\widetilde{\mathcal{T }}_0=\widetilde{\mathcal{T }}_0(\{\Gamma _n\}_{n\ge 0})$ are defined by (3.10) and (3.11), i.e.,

$$\begin{aligned} \mathcal{T }_0=P_{\mathfrak{H }_0}\mathcal{U }_0{\upharpoonright \,}\mathfrak{H }_0,\;\widetilde{\mathcal{T }}_0=P_{\widetilde{\mathfrak{H }}_0}\widetilde{\mathcal{U }}_0{\upharpoonright \,}\widetilde{\mathfrak{H }}_0. \end{aligned}$$

The operators $\mathcal{T }_0$ and $\widetilde{\mathcal{T }}_0$ are unitarily equivalent completely non-unitary contractions and Proposition 3.2 hold true. The operators given by truncated block operator CMV matrices $\mathcal{T }_m$ and $\widetilde{\mathcal{T }}_m$ obtaining from $\mathcal{U }_0$ and $\widetilde{\mathcal{U }}_0$ by deleting first $m+1$ rows and $m+1$ columns are

• co-shifts of the form

  $$\begin{aligned} \mathcal{T }_m=\widetilde{\mathcal{T }}_m=\begin{bmatrix} 0&I_{\mathfrak{D }_{\Gamma ^*_m}}&0&0&\ldots \\0&0&I_{\mathfrak{D }_{\Gamma ^*_m}}&0&\ldots \\0&0&0&I_{\mathfrak{D }_{\Gamma ^*_m}}&\ldots \\\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}:\begin{array}{l} \mathfrak{D }_{\Gamma ^*_m}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_m}\\ \oplus \\ \vdots \end{array} \rightarrow \begin{array}{l} \mathfrak{D }_{\Gamma ^*_m}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_m}\\ \oplus \\ \vdots \end{array}, \end{aligned}$$

  when $\Gamma _m$ is isometry,

• the unilateral shifts of the form

  $$\begin{aligned} \mathcal{T }_m=\widetilde{\mathcal{T }}_m=\begin{bmatrix}0&0&0&0&\ldots \\I_{\mathfrak{D }_{\Gamma _m}}&0&0&0&\ldots \\0&I_{\mathfrak{D }_{\Gamma _m}}&0&0&\ldots \\\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}:\begin{array}{l} \mathfrak{D }_{\Gamma _m}\\ \oplus \\ \mathfrak{D }_{\Gamma _m}\\ \oplus \\ \vdots \end{array}\rightarrow \begin{array}{l} \mathfrak{D }_{\Gamma _m}\\ \oplus \\ \mathfrak{D }_{\Gamma _m}\\ \oplus \\ \vdots \end{array}, \end{aligned}$$

  when $\Gamma _m$ is co-isometry.

One can see that Proposition 3.2 remains true.

The conservative systems

$$\begin{aligned} \zeta _0=\{\mathcal{U }_0;\mathfrak{M },\mathfrak{N },\mathfrak{H }_0\},\;\widetilde{\zeta }_0=\{\widetilde{\mathcal{U }}_0;\mathfrak{M },\mathfrak{N },\widetilde{\mathfrak{H }}_0\}. \end{aligned}$$

are simple and unitarily equivalent and, moreover, Theorem 3.3 remains valid.

In order to obtain precise forms of $\mathcal{U }_0, \widetilde{\mathcal{U }}_0, \mathcal{T }_0$, and $\widetilde{\mathcal{T }}_0$ one can consider the following cases:

• 1. $\Gamma _{2N}$ is isometric (co-isometric) for some $N$,

• 2. $\Gamma _{2N+1}$ is isometric (co-isometric) for some $N$,

• 3. the operator $\Gamma _{2N}$ is unitary for some $N$,

• 4. the operator $\Gamma _{2N+1}$ is unitary for some $N$.

In the following we consider all these situations and will give the forms of truncated CMV matrices. We use the sub-matrices defined by (4.3) and (4.4).

1.1 A.1 $\Gamma _{2N}$ is Isometric

Define

$$\begin{aligned} \mathfrak{H }_0&= \widetilde{\mathfrak{H }}_0=\mathfrak{D }_{\Gamma ^*_0}\bigoplus \mathfrak{D }_{\Gamma ^*_0}\bigoplus \cdots ,\; {\mathrm{if}}\quad N=0,\\ \mathfrak{H }_0&= \left( \bigoplus \limits _{n=0}^{N-1}\begin{array}{l} \mathfrak{D }_{\Gamma _{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{2n+1}} \end{array}\right) \bigoplus \mathfrak{D }_{\Gamma ^*_{2N}}\bigoplus \mathfrak{D }_{\Gamma ^*_{2N}} \bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma ^*_{2N}}\bigoplus \cdots ,\\ \widetilde{\mathfrak{H }}_0&= \left( \bigoplus \limits _{n=0}^{N-1} \begin{array}{l} \mathfrak{D }_{\Gamma ^*_{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{2n+1}} \end{array}\right) \bigoplus \mathfrak{D }_{\Gamma ^*_{2N}}\bigoplus \mathfrak{D }_{\Gamma ^*_{2N}} \bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma ^*_{2N}}\bigoplus \cdots , \quad N\ge 1. \end{aligned}$$

Define the unitary operators

$$\begin{aligned} \mathcal{M }_0&= I_{\mathfrak{M }\oplus \mathfrak{H }_0},\;\widetilde{\mathcal{M }}_0=I_{\mathfrak{N }\oplus \mathfrak{H }_0},\; N=0,\\ \mathcal{M }_0&= I_\mathfrak{M }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\bigoplus \cdots :\mathfrak{M }\bigoplus \mathfrak{H }_0\rightarrow \mathfrak{M }\bigoplus \widetilde{\mathfrak{H }}_0\\ \widetilde{\mathcal{M }}_0&= I_\mathfrak{N }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\bigoplus \cdots :\mathfrak{N }\bigoplus \mathfrak{H }_0\rightarrow \mathfrak{N }\bigoplus \widetilde{\mathfrak{H }}_0,\\&N\ge 1. \end{aligned}$$

The unitary operator $\mathcal{L }_0:\mathfrak{M }\bigoplus \widetilde{\mathfrak{H }}_0\rightarrow \mathfrak{N }\bigoplus \mathfrak{H }_0$ is defined as follows

$$\begin{aligned} \mathcal{L }_0=\left\{ \begin{array}{l} \mathbf{J}^{(r)}_{\Gamma _0}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_0}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_0}}\bigoplus \cdots ,\quad N=0,\\ \mathbf{J}_{\Gamma _0}\bigoplus \mathbf{J}^{(r)}_{\Gamma _2}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2}}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2}}}\bigoplus \cdots ,\;{\mathrm{if}}\quad N=1,\\ \mathbf{J}_{\Gamma _0}\bigoplus \left( \bigoplus \limits _{n=1}^{N-1}\mathbf{J}_{\Gamma _{2n}}\right) \bigoplus \mathbf{J}^{(r)}_{\Gamma _{2N}} \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\bigoplus \cdots ,\;{\mathrm{if}}\quad N\ge 2 \end{array}\right. . \end{aligned}$$

Define $\mathcal{U }_0=\mathcal{L }_0\mathcal{M }_0, \widetilde{\mathcal{U }}_0=\widetilde{\mathcal{M }}_0\mathcal{L }_0.$ In particular, if the operator $\Gamma _0$ is isometric, then

$$\begin{aligned} \mathcal{U }_0=\widetilde{\mathcal{U }}_0=\begin{bmatrix} \Gamma _0&I_{\mathfrak{D }_{\Gamma ^*_0}}&0&0&0&0&\ldots \\0&0&I_{\mathfrak{D }_{\Gamma ^*_0}}&0&0&0&\ldots \\0&0&0&I_{\mathfrak{D }_{\Gamma ^*_0}}&0&0&\ldots \\0&0&0&0&I_{\mathfrak{D }_{\Gamma ^*_0}}&0&\ldots \\\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}. \end{aligned}$$

The block operator truncated CMV matrices $\mathcal{T }_0$ and $\widetilde{\mathcal{T }}_0$ are the products (for $N\ge 1$):

$$\begin{aligned} \mathcal{T }_0&= \left( {-\Gamma ^*_0}\oplus \left( \bigoplus \limits _{n=1}^{N-1}\mathbf{J}_{\Gamma _{2n}}\right) \oplus \mathbf{J}^{(r)}_{\Gamma _{2N}} \oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus \cdots \right) \\&\times \left( \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus \cdots \right) ,\\ \widetilde{\mathcal{T }}_0&= \left( \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus \cdots \right) \\&\times \left( {-\Gamma ^*_0}\oplus \left( \bigoplus \limits _{n=1}^{N-1}\mathbf{J}_{\Gamma _{2n}}\right) \oplus \mathbf{J}^{(r)}_{\Gamma _{2N}} \oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus I_{\mathfrak{D }_{\Gamma ^*_{2N}}}\oplus \cdots \right) . \end{aligned}$$

Calculations give

$$\begin{aligned} \mathcal{T }_0&= \left[ \begin{array}{c|c} {\mathcal{S }_{N-1}}&{} \begin{array}{c@{\quad }c@{\quad }c} 0&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots \\ 0&{}0&{}\ldots \\ I_{\mathfrak{D }_{\Gamma ^*_{2N}}}&{}0&{}\ldots \end{array}\\ \hline \mathbf{0}&{} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N}}}&{}0&{}0&{}\ldots \\ 0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array} \end{array} \right] ,\\ \widetilde{\mathcal{T }}_0&= \left[ \begin{array}{c|c} {\widetilde{\mathcal{S }}_{N-1}}&{} \begin{array}{c@{\quad }c@{\quad }c} 0&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots \\ 0&{}0&{}\ldots \\ D_{\Gamma ^*_{2N-1}}&{}0&{}\ldots \\ -\Gamma ^*_{2N-1}&{}0&{}\ldots \end{array}\\ \hline \mathbf{0}&{} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N}}}&{}0&{}0&{}\ldots \\ 0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array} \end{array} \right] . \end{aligned}$$

1.2 A.2 $\Gamma _{2N}$ is Co-isometric

Then $\Gamma _{n}=0, \mathfrak{D }_{\Gamma _{n}}=\mathfrak{D }_{\Gamma _{2N}}, D_{\Gamma _{n}}=I_{\mathfrak{D }_{\Gamma _{2N}}}$ for $n> 2N$. Define

$$\begin{aligned} \mathfrak{H }_0&= \widetilde{\mathfrak{H }}_0=\bigoplus \limits _{n=0}^\infty \mathfrak{D }_{\Gamma _0},\; {\mathrm{if}}\quad N=0,\\ \mathfrak{H }_0&= \left( \bigoplus \limits _{n=0}^{N-1}\begin{array}{l} \mathfrak{D }_{\Gamma _{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{2n+1}} \end{array}\right) \bigoplus \mathfrak{D }_{\Gamma _{2N}}\bigoplus \mathfrak{D }_{\Gamma _{2N}} \bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma _{2N}}\bigoplus \cdots ,\\ \widetilde{\mathfrak{H }}_0&= \left( \bigoplus \limits _{n=0}^{N-1} \begin{array}{l} \mathfrak{D }_{\Gamma ^*_{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{2n+1}} \end{array}\right) \bigoplus \mathfrak{D }_{\Gamma _{2N}}\bigoplus \mathfrak{D }_{\Gamma _{2N}} \bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma _{2N}}\bigoplus \cdots ,\;{\mathrm{if}}\quad N\ge 1. \end{aligned}$$

Define the unitary operators $\mathcal{L }_0:\mathfrak{M }\bigoplus \widetilde{\mathfrak{H }}_0\rightarrow \mathfrak{N }\bigoplus \mathfrak{H }_0, \mathcal{M }_0:\mathfrak{M }\bigoplus \mathfrak{H }_0\rightarrow \mathfrak{M }\bigoplus \widetilde{\mathfrak{H }}_0$, and $\widetilde{\mathcal{M }}_0:\mathfrak{N }\bigoplus \mathfrak{H }_0\rightarrow \mathfrak{N }\bigoplus \widetilde{\mathfrak{H }}_0$ as follows

$$\begin{aligned} \mathcal{L }_0&= \mathbf{J}^{(c)}_{\Gamma _0}\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus \cdots \;{\mathrm{if}}\quad N=0,\\ \mathcal{L }_0&= \mathbf{J}_{\Gamma _0}\bigoplus \left( \bigoplus \limits _{n=0}^{N-1}\mathbf{J}_{\Gamma _{2n}}\right) \bigoplus \mathbf{J}^{(c)}_{\Gamma _{ 2N}} \bigoplus I_{\mathfrak{D }_{\Gamma _{2N}}}\bigoplus I_{\mathfrak{D }_{\Gamma _{2N}}}\bigoplus \cdots ,\;{\mathrm{if}}\quad N\ge 1,\\ \mathcal{M }_0&= I_\mathfrak{M }\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus \cdots \;{\mathrm{if}}\quad N=0,\\ \mathcal{M }_0&= I_\mathfrak{M }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus I_{\mathfrak{D }_{\Gamma _{2N}}}\bigoplus I_{\mathfrak{D }_{\Gamma _{2N}}}\bigoplus \cdots ,\;{\mathrm{if}}\quad N\ge 1, \\ \widetilde{\mathcal{M }}_0&= I_\mathfrak{N }\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus \cdots \;{\mathrm{if}}\quad N=0,\\ \widetilde{\mathcal{M }}_0&= I_\mathfrak{N }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus I_{\mathfrak{D }_{\Gamma _{2N}}}\bigoplus I_{\mathfrak{D }_{\Gamma _{2N}}}\bigoplus \cdots ,\;{\mathrm{if}}\quad N\ge 1. \end{aligned}$$

Finally define $\mathcal{U }_0=\mathcal{L }_0\mathcal{M }_0$ and $\widetilde{\mathcal{U }}_0=\widetilde{\mathcal{M }}_0\mathcal{L }_0$.

In particular, if the operator $\Gamma _{0}$ is co-isometric, then $\mathfrak{H }_0\!=\!\widetilde{\mathfrak{H }}_0\!=\!\mathfrak{D }_{\Gamma _0}\bigoplus \mathfrak{D }_{\Gamma _0}\bigoplus \cdots ,$

$$\begin{aligned} \mathcal{M }_0&= I_{\mathfrak{M }\oplus \mathfrak{H }_0},\; \widetilde{\mathcal{M }}_0=I_{\mathfrak{N }\oplus \mathfrak{H }_0},\; \mathcal{L }_0=\mathbf{J}^{(c)}_{\Gamma _0}\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus I_{\mathfrak{D }_{\Gamma _0}}\bigoplus \cdots ,\\ \mathcal{U }_0&= \widetilde{\mathcal{U }}_0=\begin{bmatrix} \Gamma _0&0&0&0&\ldots \\D_{\Gamma _0}&0&0&0&\ldots \\0&I_{\mathfrak{D }_{\Gamma _0}}&0&0&\ldots \\0&0&I_{\mathfrak{D }_{\Gamma _0}}&0&\ldots \\\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}. \end{aligned}$$

If $N\ge 1$, then truncated CMV matrices $\mathcal{T }_0$ and $\widetilde{\mathcal{T }}_0$ are of the form

$$\begin{aligned} \mathcal{T }_0&= \left[ \begin{array}{c|c} {\mathcal{S }_{N-1}}&{}\mathbf{0}\\ \hline \begin{array}{ccccc} 0&{}\ldots &{}0&{}D_{\Gamma _{2N}}D_{\Gamma _{2N-1}}&{}-D_{\Gamma _{2N}}\Gamma ^*_{2N-1} \\ 0&{}\ldots &{}0&{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}&{} \begin{array}{cccc} 0&{}0&{}0&{}\ldots \\ I_{\mathfrak{D }_{\Gamma _{2N}}}&{}0&{}0&{}\ldots \\ 0&{}I_{\mathfrak{D }_{\Gamma _{2N}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots \end{array} \end{array} \right] ,\\&\quad \widetilde{\mathcal{T }}_0=\left[ \begin{array}{c|c} {\widetilde{\mathcal{S }}_{N-1}}&{}\mathbf{0}\\ \hline \begin{array}{ccccc} 0&{}0&{}\ldots &{}0&{}D_{\Gamma _{2N}} \\ 0&{}0&{}\ldots &{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}&{} \begin{array}{cccc} 0&{}0&{}0&{}\ldots \\ I_{\mathfrak{D }_{\Gamma _{2N}}}&{}0&{}0&{}\ldots \\ 0&{}I_{\mathfrak{D }_{\Gamma _{2N}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots \end{array} \end{array} \right] . \end{aligned}$$

1.3 A.3 $\Gamma _{2N+1}$ is Isometric

In this case $\Gamma _{n}=0, \mathfrak{D }_{\Gamma ^*_{n}}=\mathfrak{D }_{\Gamma ^*_{2N+1}}, D_{\Gamma ^*_{n}}=I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}$ for $n> 2N+1$. Define

$$\begin{aligned} \small \mathfrak{H }_0&= \left( \bigoplus \limits _{n=0}^{N}\begin{array}{l} \mathfrak{D }_{\Gamma _{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{2n+1}} \end{array} \right) \bigoplus \mathfrak{D }_{\Gamma ^*_{2N+1}}\bigoplus \mathfrak{D }_{\Gamma ^*_{2N+1}}\bigoplus \ldots \bigoplus \mathfrak{D }_{\Gamma ^*_{2N+1}}\bigoplus \cdots ,\\ \widetilde{\mathfrak{H }}_0&= \mathfrak{D }_{\Gamma ^*_0}\bigoplus \mathfrak{D }_{\Gamma ^*_1} \bigoplus \mathfrak{D }_{\Gamma ^*_1}\bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma ^*_1}\bigoplus \cdots ,\quad {\mathrm{if}}\quad N=0,\\ \widetilde{\mathfrak{H }}_0\!&= \!\left( \bigoplus \limits _{n=0}^{N-1}\begin{array}{l} \mathfrak{D }_{\Gamma ^*_{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{2n+1}} \end{array}\right) \bigoplus \mathfrak{D }_{\Gamma ^*_{2N}}\bigoplus \mathfrak{D }_{\Gamma ^*_{2N+1}} \bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma ^*_{2N+1}}\bigoplus \cdots ,\quad {\mathrm{if}}\quad N{\ge } 1. \end{aligned}$$

Define the unitary operators

$$\begin{aligned} \mathcal{M }_0&={\left\{ \begin{array}{ll}I_\mathfrak{M }\bigoplus \mathbf{J}^{(r)}_{\Gamma _1} \bigoplus I _{\mathfrak{D }_{\Gamma ^*_1}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_1}}\bigoplus \cdots (N=0),\\ I_\mathfrak{M }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus \mathbf{J}^{(r)}_{\Gamma _{2N+1}} \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}} \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}\bigoplus \cdots (N\ge 1),\end{array}\right. }\\ \mathcal{L }_0&={\left\{ \begin{array}{ll}\mathbf{J}_{\Gamma _0}\bigoplus I _{\mathfrak{D }_{\Gamma ^*_1}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_1}}\bigoplus \cdots (N=0),\\ \mathbf{J}_{\Gamma _0}\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n}}\right) \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}} \bigoplus \ldots (N\ge 1),\end{array}\right. }\\ \widetilde{\mathcal{M }}_0&={\left\{ \begin{array}{ll} I_\mathfrak{N }\bigoplus \mathbf{J}^{(r)}_{\Gamma _1} \bigoplus I _{\mathfrak{D }_{\Gamma ^*_1}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_1}}\bigoplus \cdots (N=0),\\ I_\mathfrak{N }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus \mathbf{J}^{(r)}_{\Gamma _{2N+1}} \bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}\bigoplus I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}\bigoplus \cdots (N\ge 1).\end{array}\right. } \end{aligned}$$

Define $\mathcal{U }_0=\mathcal{L }_0\mathcal{M }_0$ and $\widetilde{\mathcal{U }}_0=\widetilde{\mathcal{M }}_0\mathcal{L }_0$.

If the operator $\Gamma _1$ is isometric, then

$$\begin{aligned} \mathfrak{H }_0&= \begin{array}{l} \mathfrak{D }_{\Gamma _{0}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{1}}\end{array}\bigoplus \mathfrak{D }_{\Gamma ^*_{1}}\bigoplus \mathfrak{D }_{\Gamma ^*_{1}}\bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma ^*_{1}}\bigoplus \cdots ,\\ \widetilde{\mathfrak{H }}_0&= \mathfrak{D }_{\Gamma ^*_0}\bigoplus \mathfrak{D }_{\Gamma ^*_1}\bigoplus \mathfrak{D }_{\Gamma ^*_1}\bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma ^*_1}\bigoplus \cdots ,\\ \mathcal{U }_0&= \begin{bmatrix} \Gamma _0&\quad D_{\Gamma ^*_0}\Gamma _1&\quad D_{\Gamma ^*_0}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad \ldots \\ D_{\Gamma _0}&\quad -\Gamma ^*_0\Gamma _1&\quad -\Gamma ^*_0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad \ldots \\ 0&\quad 0&\quad 0&\quad I_{\mathfrak{D }_{\Gamma ^*_1}}&\quad 0&\quad 0&\quad 0&\quad 0&\quad \ldots \\ 0&\quad 0&\quad 0&\quad 0&\quad I_{\mathfrak{D }_{\Gamma ^*_1}}&\quad 0&\quad 0&\quad 0&\quad \ldots \\ 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad I_{\mathfrak{D }_{\Gamma ^*_1}}&\quad 0&\quad 0&\quad \ldots \\ \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots \end{bmatrix},\\ \widetilde{\mathcal{U }}_0&= \begin{bmatrix}\Gamma _0&\quad D_{\Gamma ^*_0}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad \ldots \\ \Gamma _1D_{\Gamma _0}&\quad -\Gamma _1\Gamma ^*_0&\quad I_{\mathfrak{D }_{\Gamma ^*_1}}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad \ldots \\ 0&\quad 0&\quad 0&\quad I_{\mathfrak{D }_{\Gamma ^*_1}}&\quad 0&\quad 0&\quad 0&\quad 0&\quad \ldots \\ 0&\quad 0&\quad 0&\quad 0&\quad I_{\mathfrak{D }_{\Gamma ^*_1}}&\quad 0&\quad 0&\quad 0&\quad \ldots \\ \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots&\quad \vdots \end{bmatrix}. \end{aligned}$$

If $N\ge 1$, then truncated CMV matrices $\mathcal{T }_0$ and $\widetilde{\mathcal{T }}_0$ take the form

$$\begin{aligned} \mathcal{T }_0={\small \left[ \begin{array}{c|c} {\mathcal{S }_{N-1}}&{}\begin{array}{cccc} 0&{}0&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ 0&{}0&{}0&{}\ldots \\ D_{\Gamma ^*_{2N+1}}\Gamma _{2N+1}&{}D_{\Gamma ^*_{2N}}&{}0&{}\ldots \end{array}\\ \hline {\tiny \begin{array}{cccccc} 0&{}\ldots &{}0&{}D_{\Gamma _{2N}}D_{\Gamma _{2N-1}}&{}-D_{\Gamma _{2N}} \Gamma ^*_{2N-1}\\ 0&{}\ldots &{}0&{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}}&{} {\tiny \begin{array}{cccccc} -\Gamma ^*_{2N}\Gamma _{2N+1}&{}-\Gamma ^*_{2N}&{}0&{}0&{}\ldots \\ 0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}\ldots \\ 0&{}0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}} \end{array} \right] },\\ \widetilde{\mathcal{T }}_0=\left[ \begin{array}{c|c} {\widetilde{\mathcal{S }}_{N-1}}&{}{\begin{array}{cccc} 0&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots \\ 0&{}0&{}\ldots \\ D_{\Gamma ^*_{2N-1}}D_{\Gamma ^*_{2N}}&{}0&{}\ldots \\ -{\Gamma ^*_{2N-1}}D_{\Gamma ^*_{2N}}&{}0&{}\ldots \end{array}} \\ \hline \begin{array}{cccccc} 0&{}\ldots &{}0&{}\Gamma _{2N+1}D_{\Gamma _{2N}}\\ 0&{}\ldots &{}0&{}0\\ 0&{}\ldots &{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots \end{array}&{} {\small \begin{array}{cccccc} -\Gamma _{2N+1}\Gamma ^*_{2N}&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}0&{}0&{}\ldots \\ 0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}0&{}\ldots \\ 0&{}0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}} \end{array} \right] . \end{aligned}$$

1.4 A.4 $\Gamma _{2N+1}$ is Co-isometric

Then $\Gamma _{n}=0, \mathfrak{D }_{\Gamma _{n}}=\mathfrak{D }_{\Gamma _{2N+1}}$,

$D_{\Gamma _{n}}=I_{\mathfrak{D }_{\Gamma _{2N+1}}}$ for $n> 2N+1$. Define

$$\begin{aligned} \mathfrak{H }_0&= \mathfrak{D }_{\Gamma _0}\bigoplus \mathfrak{D }_{\Gamma _1}\bigoplus \mathfrak{D }_{\Gamma _1} \bigoplus \cdots ,\; {\mathrm{if}}\quad N=0,\\ \mathfrak{H }_0&= \left( \bigoplus \limits _{n=0}^{N-1}\begin{array}{l} \mathfrak{D }_{\Gamma _{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{2n+1}}\end{array}\right) \bigoplus \mathfrak{D }_{\Gamma _{2N}}\bigoplus \mathfrak{D }_{\Gamma _{2N+1}}\bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma _{2N+1}}\bigoplus \cdots \;{\mathrm{if}}\; N\ge 1,\\ \widetilde{\mathfrak{H }}_0&= \mathfrak{D }_{\Gamma ^*_0}\bigoplus \mathfrak{D }_{\Gamma _1}\bigoplus \mathfrak{D }_{\Gamma _1}\bigoplus \cdots ,\; {\mathrm{if}}\quad N=0, \\ \widetilde{\mathfrak{H }}_0&= \left( \bigoplus \limits _{n=0}^{N}\begin{array}{l}\mathfrak{D }_{\Gamma ^*_{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{2n+1}}\end{array}\right) \bigoplus \mathfrak{D }_{\Gamma _{2N+1}}\bigoplus \mathfrak{D }_{\Gamma _{2N+1}}\bigoplus \cdots \bigoplus \mathfrak{D }_{\Gamma _{2N+1}}\bigoplus \cdots \;{\mathrm{if}}\; N\ge 1. \end{aligned}$$

Define operators

$$\begin{aligned} \mathcal{L }_0&= {\left\{ \begin{array}{ll}\mathbf{J}_{D_{\Gamma _0}}\bigoplus I_{\mathfrak{D }_{\Gamma _1}}\bigoplus I_{\mathfrak{D }_{\Gamma _1}}\bigoplus \cdots \;{\mathrm{if}}\quad N=0,\\ \mathbf{J}_{\Gamma _0}\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n}}\right) \bigoplus I_{\mathfrak{D }_{\Gamma _{2N+1}}}\bigoplus I_{\mathfrak{D }_{\Gamma _{2N+1}}}\bigoplus \cdots \;{\mathrm{if}}\quad N\ge 1, \end{array}\right. }\\ \mathcal{M }_0&= {\left\{ \begin{array}{ll}I_\mathfrak{M }\bigoplus \mathbf{J}^{(c)}_{\Gamma _1}\bigoplus I_{\mathfrak{D }_{\Gamma _1}}\bigoplus I_{\mathfrak{D }_{\Gamma _1}}\bigoplus \cdots \; {\mathrm{if}}\quad N=0, \\ I_\mathfrak{M }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus \mathbf{J}^{(c)}_{\Gamma _{2N+1}} \bigoplus I_{\mathfrak{D }_{\Gamma _{2N+1}}} \bigoplus \cdots \;{\mathrm{if}}\quad N\ge 1, \end{array}\right. }\\ \widetilde{\mathcal{M }}_0&= {\left\{ \begin{array}{ll}I_\mathfrak{N }\bigoplus \mathbf{J}^{(c)}_{\Gamma _1}\bigoplus I_{\mathfrak{D }_{\Gamma _1}}\bigoplus I_{\mathfrak{D }_{\Gamma _1}}\bigoplus \cdots \; {\mathrm{if}}\quad N=0, \\ I_\mathfrak{N }\bigoplus \left( \bigoplus \limits _{n=1}^{N}\mathbf{J}_{\Gamma _{2n-1}}\right) \bigoplus \mathbf{J}^{(c)}_{\Gamma _{2N+1}} \bigoplus I_{\mathfrak{D }_{\Gamma _{2N+1}}}\bigoplus \cdots \;{\mathrm{if}}\quad N\ge 1. \end{array}\right. } \end{aligned}$$

Define $\mathcal{U }_0=\mathcal{L }_0\mathcal{M }_0$ and $\widetilde{\mathcal{U }}_0=\widetilde{\mathcal{M }}_0\mathcal{L }_0$.

If $\Gamma _1$ is co-isometric, then

$$\begin{aligned} \mathcal{U }_0=\begin{bmatrix} \Gamma _0&D_{\Gamma ^*_0}\Gamma _1&0&0&0&\ldots \\D_{\Gamma _0}&-\Gamma ^*_0\Gamma _1&0&0&0&\ldots \\0&{D_{\Gamma _1}}&0&0&0&\ldots \\0&0&I_{\mathfrak{D }_{\Gamma _1}}&0&0&\ldots \\0&0&0&I_{\mathfrak{D }_{\Gamma _1}}&0&\ldots \\\vdots&\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix},\quad \widetilde{\mathcal{U }}_0=\begin{bmatrix} \Gamma _0&D_{\Gamma ^*_0}&0&0&0&\ldots \\\Gamma _1D_{\Gamma _0}&-\Gamma _1\Gamma ^*_0&0&0&0&\ldots \\D_{\Gamma _1}D_{\Gamma _0}&-{D_{\Gamma _1}}\Gamma ^*_0&0&0&0&\ldots \\0&0&I_{\mathfrak{D }_{\Gamma _1}}&0&0&\ldots \\0&0&0&I_{\mathfrak{D }_{\Gamma _1}}&0&\ldots \\\vdots&\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}. \end{aligned}$$

If $N\ge 1$, then truncated CMV matrices $\mathcal{T }_0$ and $\widetilde{\mathcal{T }}_0$ in this case take the form

$$\begin{aligned} \mathcal{T }_0&= \left[ \begin{array}{c|c} {\mathcal{S }_{N-1}}&{}\begin{array}{cccccc} 0&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots \\ 0&{}0&{}\ldots \\ D_{\Gamma ^*_{2N}}\Gamma _{2N+1}&{}0&{}\ldots \end{array} \\ \hline \begin{array}{ccccc} 0&{}\ldots &{}0&{}D_{\Gamma _{2N}}D_{\Gamma _{2N-1}}&{}-D_{\Gamma _{2N}}\Gamma ^*_{2N-1}\\ 0&{}\ldots &{}0&{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}&{} \begin{array}{cccccc} -\Gamma ^*_{2N}\Gamma _{2N+1}&{}0&{}0&{}\ldots \\ D_{\Gamma _{2N+1}}&{}0&{}0&{}\ldots \\ 0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array} \end{array} \right] ,\\ \widetilde{\mathcal{T }}_0&= \left[ \begin{array}{c|c} {\widetilde{\mathcal{S }}_{N-1}}&{}\begin{array}{cccccc} 0&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots \\ 0&{}0&{}\ldots \\ D_{\Gamma ^*_{2N-1}}D_{\Gamma ^*_{2N}}&{}0&{}\ldots \\ -{\Gamma ^*_{2N-1}}D_{\Gamma ^*_{2N}}&{}0&{}\ldots \end{array} \\ \hline {\begin{array}{ccccc} 0&{}\ldots &{}0&{}\Gamma _{2N+1}D_{\Gamma _{2N}}\\ 0&{}\ldots &{}0&{}D_{\Gamma _{2N+1}}D_{\Gamma _{2N}}\\ 0&{}\ldots &{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots \end{array}}&{} {\begin{array}{cccccc} -\Gamma _{2N+1}\Gamma ^*_{2N}&{}0&{}0&{}\ldots \\ -D_{\Gamma _{2N+1}}\Gamma ^*_{2N}&{}0&{}0&{}\ldots \\ 0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}0&{}\ldots \\ 0&{}0&{}I_{\mathfrak{D }_{\Gamma ^*_{2N+1}}}&{}0&{}\ldots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \end{array}} \end{array} \right] . \end{aligned}$$

1.5 A.5 $\Gamma _{2N}$ is Unitary

In this case

$$\begin{aligned} \mathfrak{H }_0&= \bigoplus \limits _{n=0}^{N-1}\begin{array}{l} \mathfrak{D }_{\Gamma _{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{2n+1}} \end{array},\; \widetilde{\mathfrak{H }}_0= \bigoplus \limits _{n=0}^{N-1}\begin{array}{l} \mathfrak{D }_{\Gamma ^*_{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{2n+1}}\end{array},\\ \mathcal{U }_0&= \left( \mathbf{J }_{\Gamma _0}\oplus \mathbf{J }_{\Gamma _2}\oplus \cdots \oplus \mathbf{J }_{\Gamma _{2(N-1)}}\oplus \Gamma _{2N}\right) \times \left( I_{\mathfrak{M }}\oplus \mathbf{J }_{\Gamma _1}\oplus \cdots \oplus \mathbf{J }_{\Gamma _{2N-1}}\right) , \end{aligned}$$

If $N=1$ ($\Gamma _2$ is unitary), then

$$\begin{aligned} \mathcal{U }_0=\begin{bmatrix}\Gamma _0&\quad D_{\Gamma ^*_0}\Gamma _1&\quad D_{\Gamma ^*_0}D_{\Gamma ^*_1}\\ D_{\Gamma _0}&\quad -\Gamma ^*_0\Gamma _1&\quad -\Gamma ^*_0D_{\Gamma ^*_1}\\ 0&\quad \Gamma _2 D_{\Gamma _1}&\quad -\Gamma _2\Gamma ^*_1 \end{bmatrix},\;\widetilde{\mathcal{U }}_0=\begin{bmatrix}\Gamma _0&\quad D_{\Gamma ^*_0}&\quad 0&\\ \Gamma _1D_{\Gamma _0}&\quad -\Gamma _1\Gamma ^*_0&\quad D_{\Gamma ^*_1}\Gamma _2\\ D_{\Gamma _1}D_{\Gamma _0}&\quad -D_{\Gamma _1}\Gamma ^*_0&\quad -\Gamma ^*_1\Gamma _2 \end{bmatrix} \end{aligned}$$

If $N\ge 1$, then

$$\begin{aligned} \mathcal{T }_0= \mathcal{S }_{N-1},\; \widetilde{\mathcal{T }}_0= \widetilde{\mathcal{S }}_{N-1}. \end{aligned}$$

1.6 A.6 $\Gamma _{2N+1}$ is Unitary

Then

$$\begin{aligned} \mathfrak{H }_0&= \mathfrak{D }_{\Gamma _0},\;\widetilde{\mathfrak{H }}_0=\mathfrak{D }_{\Gamma ^*_0}\;{\mathrm{if}}\quad N=0,\\ \mathfrak{H }_0&= \bigoplus \limits _{n=0}^{N-1}\begin{array}{l}\mathfrak{D }_{\Gamma _{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma ^*_{2n+1}}\end{array}\bigoplus \mathfrak{D }_{\Gamma _{2N}},\; \widetilde{\mathfrak{H }}_0= \bigoplus \limits _{n=0}^{N-1}\begin{array}{l}\mathfrak{D }_{\Gamma ^*_{2n}}\\ \oplus \\ \mathfrak{D }_{\Gamma _{2n+1}}\end{array}\bigoplus \mathfrak{D }_{\Gamma ^*_{2N}}\;{\mathrm{if}}\; N\ge 1,\\ \mathcal{U }_0&= \left( \mathbf{J }_{\Gamma _0}\oplus \mathbf{J }_{\Gamma _2}\oplus \cdots \oplus \mathbf{J }_{\Gamma _{2N}}\right) \times \left( I_{\mathfrak{M }}\oplus \mathbf{J }_{\Gamma _1}\oplus \cdots \oplus \mathbf{J }_{\Gamma _{2N-1}}\oplus \Gamma _{2N+1}\right) ,\\ \widetilde{\mathcal{U }}_0&= \left( I_{\mathfrak{N }}\oplus \mathbf{J }_{\Gamma _1}\oplus \cdots \oplus \mathbf{J }_{\Gamma _{2N-1}}\oplus \Gamma _{2N+1}\right) \times \left( \mathbf{J }_{\Gamma _0}\oplus \mathbf{J }_{\Gamma _2}\oplus \cdots \oplus \mathbf{J }_{\Gamma _{2N}}\right) , \quad N\ge 1.\\ \mathcal{U }_0&= \begin{bmatrix}\Gamma _0&D_{\Gamma ^*_0}\\D_{\Gamma _0}&-\Gamma ^*_0\end{bmatrix}\begin{bmatrix}I_\mathfrak{M }&0\\0&\Gamma _1 \end{bmatrix}=\begin{bmatrix}\Gamma _0&D_{\Gamma ^*_0}\Gamma _1\\D_{\Gamma _0}&-\Gamma ^*_0\Gamma _1 \end{bmatrix},\\ \widetilde{\mathcal{U }}_0&= \begin{bmatrix}I_\mathfrak{N }&0\\0&\Gamma _1 \end{bmatrix}\begin{bmatrix}\Gamma _0&D_{\Gamma ^*_0}\\D_{\Gamma _0}&-\Gamma ^*_0\end{bmatrix}=\begin{bmatrix}\Gamma _0&D_{\Gamma ^*_0}\\\Gamma _1D_{\Gamma _0}&-\Gamma _1\Gamma ^*_0 \end{bmatrix},\;{\mathrm{if}}\quad N=0, \end{aligned}$$

In this case if $N\ge 1$, then

$$\begin{aligned} \mathcal{T }_0&= \left[ \begin{array}{c|c} \mathcal{S }_{N-1} &{} \begin{array}{cc} 0\\ 0\\ \vdots \\ 0\\ D_{\Gamma ^*_{2N}}\Gamma _{2N+1} \end{array}\\ \hline \begin{array}{ccccc} 0&{}\ldots &{}0&{}D_{\Gamma _{2N}}D_{\Gamma _{2N-1}}&{}-D_{\Gamma _{2N}}{\Gamma ^*_{2N-1}} \end{array}&\begin{array}{cc} -\Gamma ^*_{2N}\Gamma ^*_{2N+1} \end{array} \end{array} \right] ,\\ \widetilde{\mathcal{T }}_0&= \left[ \begin{array}{c|c} \widetilde{\mathcal{S }}_{N-1} &{} \begin{array}{cc} 0\\ 0\\ \vdots \\ 0\\ D_{\Gamma ^*_{2N-1}}D_{\Gamma ^*_{2N}}\\ -\Gamma _{2N-1}D_{\Gamma ^*_{2N}} \end{array}\\ \hline \begin{array}{ccccc} 0&{}\ldots &{}0&{}{\Gamma _{2N+1}}D_{\Gamma _{2N}} \end{array}&\begin{array}{cc} -\Gamma _{2N+1}\Gamma ^*_{2N} \end{array} \end{array} \right] . \end{aligned}$$

In particular if $N=1$ ($\Gamma _3$ is unitary), then

$$\begin{aligned} \mathcal{U }_0=\begin{bmatrix}\Gamma _0&D_{\Gamma ^*_0}\Gamma _1&D_{\Gamma ^*_0}D_{\Gamma ^*_1}&0\\ D_{\Gamma _0}&-\Gamma ^*_0\Gamma _1&-\Gamma ^*_0D_{\Gamma ^*_1}&0\\ 0&\Gamma _2 D_{\Gamma _1}&-\Gamma _2\Gamma ^*_1&D_{\Gamma ^*_2}\Gamma _3\\ 0&D_{\Gamma _2}D_{\Gamma _1}&-D_{\Gamma _2}\Gamma ^*_1&-\Gamma ^*_2\Gamma _3 \end{bmatrix},\;\widetilde{\mathcal{U }}_0=\begin{bmatrix}\Gamma _0&D_{\Gamma ^*_0}&0&0&\\ \Gamma _1D_{\Gamma _0}&-\Gamma _1\Gamma ^*_0&D_{\Gamma ^*_1}\Gamma _2&D_{\Gamma ^*_1}D_{\Gamma ^*_2}\\ D_{\Gamma _1}D_{\Gamma _0}&-D_{\Gamma _1}\Gamma ^*_0&-\Gamma ^*_1\Gamma _2&-\Gamma ^*_1D_{\Gamma ^*_2}\\ 0&0&\Gamma _3 D_{\Gamma _2}&-\Gamma _3\Gamma ^*_2 \end{bmatrix}. \end{aligned}$$