# V.V. Napalkov, V.V. Napalkov, Jr. On the coincidence of reproducing kernel Hilbert spaces connected by a special transformation ... P. 149-159

We consider two reproducing kernel Hilbert spaces $H_1$ and $H_2$ consisting of complex-valued functions given on some sets $\Omega_1\subset {\mathbb C}^n$ and $\Omega_2\subset {\mathbb C}^m$, respectively. The norms in $H_1$ and $H_2$ have integral form:
$$\| f\|_{H_1}^2=\int_ {\Omega_1}|f (z)|^2\, d\mu(z), \ \ f\in H_1;\ \ \ \ \ \| q\|_{H_2}^2=\int_{\Omega_2}|q(t)|^2\,d\nu(t), \ \ q\in H_2.$$
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{\rm def}{=}(E(\cdot, z), f)_{H_1}\   \forall z\in \Omega_2,\ \  \widetilde H_1=\{\widetilde f,\, f\in H_1\},
(\widetilde f_1,\widetilde f_2)_{\widetilde H_1}\stackrel{\rm def}{=}(f_2,f_1)_{H_1},
\|\widetilde f_1\|_{\widetilde H_1}=\|f_1\|_{H_1}\ \ \forall \widetilde  f_1,\widetilde f_2\in \widetilde H_1.
\end{align*}
We study the question of coincidence of the spaces $\widetilde H_1$ and $H_2$, i.e., the conditions under which these spaces consist of the same functions and have equal norms. The following criterion of coincidence is obtained: $\widetilde H_1=H_2$ if and only if there exists a linear continuous one-to-one unitary operator ${\cal A}$ from $\overline H_1$ onto $H_2$ that for any $\xi\in \Omega_1$ takes the function $K_{\overline H_1}(\cdot,\xi)$ to the function $E(\xi,\cdot)$. Here $\overline H_1$ is the space consisting of the complex conjugates of functions from $H_1$ and $K_{\overline H_1}(t,\xi)$, $t,\xi\in \Omega_1$, is the reproducing kernel of the space $\overline H_1$. We also obtain some equivalent statements and a criterion for the coincidence of $H_1$ and $H_2$.

Keywords: Bargmann-Fock space, operator of multiplication by a function, expansion systems similar to orthogonal systems, reproducing kernel Hilbert space

Revised March 27, 2019

Accepted April 29, 2019

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 17-41-020070).

Valentin Vasilievich Napalkov, Dr. Phys.-Math. Sci., Prof., Corresponding Member of RAS, Institute of Mathematics, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, 450077 Russia, e-mail: napalkov@matem.anrb.ru

Valerii Valentinovich Napalkov, Cand. Sci. (Phys.-Math.), Institute of Mathematics, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, 450077 Russia, e-mail: vnap@mail.ru

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Cite this article as: V.V.Napalkov, V.V.Napalkov (Jr.). On the coincidence of reproducing kernel Hilbert spaces connected by a special transformation, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 149–159 .