On the equivalence of reproducing kernel Hilbert spaces connected by a special transform
We consider two reproducing kernel Hilbert spaces $H_1$ and $H_2$ consisting of complex-valued functions defined on some sets of points $\Omega_1\subset {\mathbb C}^n$ and $\Omega_2\subset {\mathbb C}^m$, respectively. The norms in the spaces $H_1$ and $H_2$ have an integral form: $\|f\|_{H_1}^2=\int_{\Omega_1}|f(t)|^2\,d\mu_1(t), \ \ f\in H_1,\quad \|q\|_{H_2}^2=\int_{\Omega_2}|q(z)|^2\,d\mu_2(z), \ \ q\in H_2.$  Let $\{E (\cdot, z)\}_{z\in \Omega_2}$ be some complete system of functions in the space $H_1$. Define $\begin{align*} \widetilde f(z)\stackrel{def}{=}(E(\cdot, z), f)_{H_1} \ \ \forall z\in \Omega_2,\quad \widetilde H_1=\{\widetilde f,\, f\in H_1\}, \\ (\widetilde f_1,\widetilde f_2)_{\widetilde H_1}\stackrel{def}{=}(f_2,f_1)_{H_1}, \quad \|\widetilde f_1\|_{\widetilde H_1}=\|f_1\|_{H_1} \ \ \forall\,\widetilde f_1,\,\widetilde f_2\in \widetilde H_1. \end{align*}$  We prove that the Hilbert spaces $\widetilde H_1$ and $H_2$ are equivalent (i.e., consist of the same functions and have equivalent norms) if and only if there exists a linear continuous one-to-one operator ${\cal A}$ acting from the space $\overline H_1$ onto the space~$H_2$ that for any $\xi\in \Omega_1$ takes the function $K_{\overline H_1}(\cdot,\xi)$ to the function $E(\xi,\cdot)$, where $\overline H_1$ is the space consisting of functions that are complex conjugate to functions from $H_1$ and $K_{\overline H_1}(t,\xi)$, $t,\xi\in \Omega_1$, is the reproducing kernel of $\overline H_1$. We also obtain other conditions for the equivalence of the spaces $\widetilde H_1$ and $H_2$. In addition, we study the question of the equivalence of the spaces $\check H_2$ and~$H_1$ and the question of the existence of special orthosimilar expansion systems in the spaces $H_1$ and $H_2$. We derive a necessary and sufficient condition for the equivalence of the spaces $H_1$ and $H_2$. This paper continues the authors' paper in which the case of coinciding spaces $\widetilde H_1$ and $H_2$ was considered.