In a section of a book that I'm reading about Hardy spaces, the author writes
$\mathcal H_2$ is a (closed) subspace of $\mathcal L_2(j \mathbb R)$ with matrix functions $F(s)$ analytic in $\text{Re}(s) > 0$ (open right-half plane). The corresponding norm is defined as $$\| F \|_2^2 = \sup_{\sigma > 0} \left\{\frac{1}{2\pi} \int_{-\infty}^\infty \text{trace}(F^*(\sigma + j\omega)F(\sigma + j\omega)) \, \text{d}\omega\right\} \label{eq:H2} \tag{1}$$
$\mathcal L_2 (j\mathbb R)$ was earlier defined as
$\mathcal L_2(j \mathbb R)$ or simply $\mathcal L_2$ is a Hilbert space of matrix-valued (or scalar-valued) functions on $j\mathbb R$ and consists of all complex matrix functions $F$ such that the following integral is bounded: $$\int_{-\infty}^\infty \text{trace}\left[F^*(j\omega)F(j\omega)\right] \, \text{d}\omega < \infty$$
I'm a bit confused about this. The author first defined $\mathcal L_2(j\mathbb R)$ to be the set of square-integrable functions $F : j\mathbb R \to \mathbb C^{m \times n}$, and then goes on to write that $\mathcal H_2$ is a subspace of $\mathcal L_2 (j \mathbb R)$. This implies that the domain for each function in $\mathcal H_2$ is $j \mathbb R$. However, in $(1)$ above, we see that the norm for $\mathcal H_2$ is defined using the expression $F(\sigma + j \omega)$, which implies that $F$ takes in inputs in the open right-half of the complex plane, even though its domain should only be the imaginary axis. What am I missing?