Preliminaries
Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$.
These random functions are drawn from a Gaussian measure $\gamma$, having zero mean and covariance operator $\mathcal{C} = -\big(\frac{d^2}{dx^2}\big)^{-\alpha}$ for $\alpha > 0$, where $\mathcal{C}$ encodes the $2\pi$-periodic boundary conditions.
We define the mean of the field as $$ m = \int_H x\ d\gamma(x) \qquad (\text{in our case}\ m \equiv 0)$$ and the covariance as the bilinear form $H^* \times H^* \to \mathbb{R}$ (however, $H^* = H$ by Riesz representation thm.) $$ C(h^*,k^*) = \int_H h^*(x)k^*(x)\ d\gamma(x) = \int_H \langle h,x\rangle \langle k,x \rangle\ d\gamma(x)$$ where $\langle\cdot,\cdot\rangle$ denotes the standard inner product in $L^2$. We furthermore define the covariance operator $\mathcal{C}:H^* \to H$ by $$ \mathcal{C} h^* = \int_H x \langle h,x \rangle\ d\gamma(x)\quad \text{and see that}\quad \langle \mathcal{C} h^*,k^* \rangle_{H\times H^*} = C(h^*,k^*)$$ In the end, we introduce the norm defined by the inner product $$ \| h \|^2_\gamma = \langle h,h\rangle_{\gamma} = \int_H \langle h,x \rangle\langle h,x \rangle\ d\gamma(x) = \langle \mathcal{C} h,h \rangle \qquad (\text{variance of}\ h)$$
Then, the space $(H_\gamma,\|\cdot \|_\gamma)$ is called the Cameron-Martin space of the measure $\gamma$.
Question
We see that for the norm of $H_\gamma$ it holds that $$ \| h \|_\gamma^2 = \int_H \langle h,x \rangle\langle h,x \rangle\ d\gamma(x) = \langle \mathcal{C}h,h\rangle = \langle -\big(\frac{d^2}{dx^2}\big)^{-\alpha}h,h\rangle$$ so that we can compute a simpler Lebesgue integral instead of the Bochner integral under the measure $\gamma$.
To convince myself, I numerically computed the Bochner integral by sampling and saw that it converges to the Lebesgue integral.
Now, what I don't understand is the relation that M. Hairer (Exercice 3.34) and others (Bogachev etc.) state which reads
$$ \boxed{\langle h,k \rangle_\gamma = \langle \mathcal{C}^{-1/2}h,\mathcal{C}^{-1/2}k\rangle }$$
If I pick $h$ and $k$ and compute $\int_H \langle h,x \rangle\langle k,x \rangle\ d\gamma(x)$ I see that it equals $\langle \mathcal{C}h,k\rangle$ and not $\langle \mathcal{C}^{-1/2}h,\mathcal{C}^{-1/2}k\rangle$.
Furthermore, I see that $H_\gamma = \mathcal{C}(H)$ and not $H_\gamma = \sqrt{\mathcal{C}}(H)$ as it is often stated in the literature.
I'd really appreciate any help! Thanks in advance!