Let $T: L^2(\mathbb R)\to O(\mathbb C)$ an transformation from the Hilbert space $L^2(\mathbb R)$ to the Fréchet spaces $O(\mathbb C)$ of all holomorphic functions on $\mathbb C$. We assume that $T$ is injective and continuous.
Now, Can we endowed the space $\mathcal H:= Im T$ (image of $T$) with the Hilbert inner product from $\left(L^{2}(\mathbb{R}), \left<.,.\right>\right)$ for that $\mathcal H:= Im T$ becomes a Hilbert space of holomorphic functions ? If yes, how?
I think it is possible if we consider the inner product $(f,g)_{\mathcal H}=\left<T^{-1}(f),T^{-1}(g)\right>_{L^{2}(\mathbb{R})}$ ??
Thank you in advance