Z-transform and$ H_2$ space

Question

The following is from the preliminaries of a paper.

Let $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ be the unit disc of complex numbers. A function
$G:(\mathbb{C} \cup \{\infty\})\backslash \mathbb{D} \rightarrow \mathbb{C}^{p \times q}$ is in $H_2$ if it can be expanded as:

$G(z) = \sum_{i=0}^\infty \frac{1}{z^i}G_i$, where $G_i \in \mathbb{C}^{q \times q}$ and $\sum_{i=0}^\infty \text Tr\ (G_iG_i^T) < \infty $

My problem is what is the relation between this expansion and the requirement to be in $H_2$ space?
(I know the definition of $H_2$ space.)

Answer 1

Let me answer my question:

Notation:

1. $G_i$ is the impulse response of the system
2. $G(z)$ is the frequency domain ($z$-domain for discrete time case)

According to the definition, if $G(z)$ is in $H_2$, its $H_2$-norm must be bounded. So we obtain the following:

1. $\sqrt {\frac{1}{2\pi}\oint tr(G(z)^HG(z))dz}<\infty$
   $z = e^{j\theta}$

By Parseval's theorem, we obtain

2. $\sqrt{\sum_{i=0}^\infty tr(G_iG_i^T)} < \infty$

Note here that, we assume the transfer matrix of impluse response are real.

Reference:
1. Book: Model Fitting in Frequency Domain Imposing Stability of the Model
2. Book: Multivariable Feedback Control Analysis and Design