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{\mathrm{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{\mathrm{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{\mathrm{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{\mathrm{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).
Keywords: orthosimilar decomposition systems, reproducing kernel Hilbert space, Riesz basis, problem of describing the dual space
Received April 28, 2022
Revised August 10, 2022
Accepted August 15, 2022
Valerii Valentinovich Napalkov, Dr. Phys.-Math. Sci., Institute of Mathematics, Ufa Federal Research Centre RAS, Ufa, 450077 Russia, e-mail: vnap@mail.ru
Andrey Alexandrovich Nuyatov, Cand. Sci. (Phys.-Math.), Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603950 Russia, e-mail: nuyatov1aa@rambler.ru
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Cite this article as: V.V. Napalkov (jr.), A.A. Nuyatov. On a condition for the coincidence of transform spaces for functionals in a Hilbert space. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 142–154.
