Question
I'm reading a paper (p.4) on linear systems where the authors silently introduce the notation for the set
$\frac{1}{z}\mathcal{RH}_\infty$.
Unfortunately I don't get the meaning of the notation nor can't I even describe in words what this is meant to be (so I don't know what to Google for...). Can you help me on that?
What I know / assume:
$\mathcal{H}_2$ and $\mathcal{H}_\infty$ are Hardy Spaces for functions with bounded 2- / respectively infinity-norm, and (as described in the paper) $\mathcal{RH}_2$ and $\mathcal{RH}_\infty$ are the set of real-rational strictly proper / proper transfer matrices. This still makes sense to me.
The paper covers discrete time systems, so I guess that $\frac{1}{z}$ is about an additional property in the z-domain, but what exactly?
I know the following definitions of the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ norms (taken from "Multivariable Feedback Control" by Skogestad) for continuous-time systems:
If $G(s)$ is a (matrix valued) transfer function, then
$\|G(s)\|_2 = \left( \frac{1}{2\pi} \int_{-\infty}^{\infty} \|G(j\omega)\|^2_F d\omega \right)^{\frac{1}{2}}$ where $\|G(j\omega)\|^2_F = \sum_{ij} |G_{ij}(j\omega)|^2$ is the Frobenius norm
and
$\|G(s)\|_\infty = \max_\omega \bar\sigma (G(j\omega))$ where $\bar\sigma (G(j\omega))$ is the largest singular value of $G(s)$ at $s=j\omega$.
(The above definitions are generalizations of the scalar valued definitions $\|f(s)\|_2 = \left( \frac{1}{2\pi} \int_{-\infty}^{\infty} |f(j\omega)|^2d\omega \right)^{\frac{1}{2}}$ and $\|f(s)\|_\infty = \max_\omega |f(s)| = \lim_{p\to\infty} \left( \int_{-\infty}^{\infty} |f(j\omega)|^pd\omega \right)^{\frac{1}{p}}$).
Keywords for text search: H2, H inf, H infinity, H$\infty$, RH2, 1/z RH inf, 1/z RH infinity, 1/z RH$\infty$
Thanks to @calvin-khor and a supervisor I have an answer to it now.
TL;DR
$\frac{1}{z}\mathcal{RH}_\infty$ denotes the set of discrete time real proper transfer function matrices which have been delayed by one timestep and thus it's simply the set of discrete time real strictly proper transfer function matrices.
Background
A transfer function being proper means that it can be expressed as a fraction of two polynomials where the degree of the denominator polynomial's degree is greater or equal than the numerator polynomial's.
Such a transfer function can be written as the sum of a subsystem that is given by a differential equation (continuous time) or difference equation (discrete time) and a direct feedthrough term. This means that the impulse response can start earliest at $t=0$, i.e. $g(t) = 0 \quad \forall t<0$ (CT) respectively $g[n]=0 \quad \forall n<0$ (DT).
Similarly, strictly proper transfer functions have $\deg(\text{denominator})>\deg(\text{numerator})$ and thus lack the feedthrough term. For a DT system this means that the impulse response can start earliest at $n=1$.
A property of the z-Transformation is that a time shift, expressed as $f[n-k]$ (here a delay if $k$ positive), translates into $z^{-k}F(z)$ in the z-domain. So a factor of $\frac{1}{z}$ delays the whole impulse response by $1$.
(But this now begs the question - why not simply write $\mathcal{RH}_2$? I suspect the authors of the paper still see a minor difference between the two sets.)