On a condition for the coincidence of transform spaces for functionals in a Hilbert space

The paper considers the following problem. Let $H$ be some reproducing kernel Hilbert space consisting of functions given on a set $\Omega\subset {\mathbb C}^n$, $n\ge1$, and let $\{e_1(\cdot,\xi)\}_{\xi\in \Omega_1}$ and $\{e_2(\cdot,\xi)\}_{\xi\in\Omega_1}$ be some complete systems of functions in $H$, where $\Omega_1\subset {\mathbb C^m}$, $m\ge1$. Define

\begin{align*} \widetilde f(z)\stackrel{def}{=}(e_1(\cdot, z), f)_{H}\, \forall z\in \Omega_1,\quad \widetilde H=\{\widetilde f,\, f\in H\}, \\ (\widetilde f_1,\widetilde f_2)_{\widetilde H}\stackrel{def}{=}(f_2,f_1)_{H}, \, \|\widetilde f_1\|_{\widetilde H}=\|f_1\|_{H} \quad\forall \widetilde f_1,\widetilde f_2\in \widetilde H, \\ \widehat f(z)\stackrel{def}{=}(e_2(\cdot, z), f)_{H}\, \forall z\in \Omega_1,\quad \widehat H=\{\widehat f,\, f\in H\}, \\ (\widehat f_1,\widehat f_2)_{\widehat H}\stackrel{def}{=}(f_2,f_1)_{H}, \, \|\widehat f_1\|_{\widehat H}=\|f_1\|_{H} \quad\forall \widehat f_1,\widehat f_2\in \widehat H. \end{align*}

It is required to find a condition under which the spaces $\widehat H$ and $\widetilde H$ coincide, i.e., $\widehat H$ and $\widetilde H$ consist of the same functions and \[ \|f\|_{\widehat H}=\|f\|_{\widetilde H} \forall f\in \widehat H=\widetilde H. \] We also study the question of conditions under which the spaces $\widehat H$ and $\widetilde H$ are equivalent. In the case when the systems of functions $\{e_j(\cdot,\xi)\}_{\xi\in\Omega_1}$, $j=1,2$, are orthosimilar decomposition systems in the space $H$ with the same measure $\mu$ given on $\Omega_1$, a criterion is established; more exactly, a condition is found that is necessary and sufficient for the coincidence (equivalence) of the spaces $\widehat H$ and $\widetilde H$. Note that, in the case of an arbitrary space $H$ and arbitrary systems of functions $\{e_1(\cdot,\xi)\}_{\xi\in \Omega_1}$ and $\{e_2(\cdot,\xi)\}_{\xi\in \Omega_1}$ that are complete in $H$, the found condition is always necessary; i.e., if the spaces $\widehat H$ and $\widetilde H$ coincide (are equivalent), then this condition is fulfilled. In the case when the systems of functions $\{e_1(\cdot,\xi)\}_{\xi\in \Omega_1}$ and $\{e_2(\cdot,\xi)\}_{\xi\in \Omega_1}$ are orthosimilar decomposition systems in the space $H$ with different measures $\mu_1$ and $\mu_2$, respectively, given on $\Omega_1$, we construct specific examples of spaces $H$ and systems of functions $\{e_1(\cdot,\xi)\}_{\xi\in \Omega_1}$ and $\{e_2(\cdot,\xi)\}_{\xi\in \Omega_1}$ complete in $H$ and such that the specified condition is met, but the spaces $\widehat H$ and $\widetilde H$ are not the same (not equivalent).