# V.V. Napalkov (jr.), A.A. Nuyatov. On a condition for the coincidence of transform spaces for functionals in a Hilbert space ... P. 142-154

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

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

REFERENCES

1.   Halmos P.R. A Hilbert space problem book. NY,: Springer, 1982, 373 p. doi: 10.1007/978-1-4684-9330-6 . Translated to Russian under the title Gil’bertovo prostranstvo v zadachakh, Moscow: Mir Publ., 1970, 351 p.

2.   Lukashenko T.P. Properties of expansion systems similar to orthogonal ones. Izv. Math., 1998, vol. 62, no. 5, pp. 1035–1054. doi: 10.1070/IM1998v062n05ABEH000215

3.   Napalkov V.V., Napalkov V.V. (jr.) On isomorphism of reproducing kernel Hilbert spaces. Dokl. Math., 2017, vol. 95, no. 3, pp. 270–272. doi: 10.1134/S1064562417030243

4.   Kantorovich L.V., Akilov G.P. Functional analysis, 2nd ed. Pergamon, 1982, 604 p. ISBN: 9781483138251 . Original Russian text published in Kantorovich L.V., Akilov G.P. Funktsional’nyi analiz, Moscow: Nauka Publ., 1984, 752 p.

5.   Levin B.Ya., Lyubarskii Yu.I. Interpolation by means of special classes of entire functions and related expansions in series of exponentials. Math. USSR Izv., 1975, vol. 9, no. 3, pp. 621–662. doi: 10.1070/IM1975v009n03ABEH001493

6.   Napalkov V.V. (jr.) On orthosimilar systems in a space of analytical functions and the problem of describing the dual space. Ufa Math. J., 2011, vol. 3, no. 1, pp. 30–41.

7.   Isaev K.P. Bezuslovnye bazisy iz eksponent v prostranstvah Bergmana na vypuklyh oblastyah [Unconditional bases of exponentials in Bergman spaces on convex domains]. Dissertation, Cand. Sci. (Phys.–Math.), Ufa: IMVTs UNTs RAN, 2004, 173 p.

8.   Isaev K.P., Yulmukhametov R.S. The absence of unconditional bases of exponentials in Bergman spaces on non-polygonal domains. Izv. Math., 2007, vol. 71, no. 6, pp. 1145–1166. doi: 10.1070/IM2007v071n06ABEH002385

9.   Gaier D. Lectures on complex approximation. Boston: Birkhauser, 1987, 196 p. doi: 10.1007/978-1-4612-4814-9 . Translated to Russian under the title Lektsii po teorii approksimatsii v kompleksnoi oblasti. Moscow: Mir Publ., 1986, 216 p.

10.   Napalkov V.V., Napalkov V.V. (jr.) On the equivalence of reproducing kernel Hilbert spaces connected by a special transform. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 200–215 (in Russian). doi: 10.21538/0134-4889-2019-25-2-149-159

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.