V.V. Napalkov, V.V. Napalkov,  Jr.  On the equivalence of reproducing kernel Hilbert spaces connected by a special transform ... P. 200-215

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:
\begin{align*}
\|f\|_{H_1}^2=\int_{\Omega_1}|f(t)|^2\,d\mu_1(t), \ \ f\in H_1,\qquad \|q\|_{H_2}^2=\int_{\Omega_2}|q(z)|^2\,d\mu_2(z), \ \ q\in H_2.
\end{align*}
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.

Keywords: orthosimilar decomposition systems, reproducing kernel Hilbert space, problem of describing the dual  space

Received February 5, 2020

Revised May 13, 2020

Accepted May 18, 2020

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

Valerii Valentinovich Napalkov Cand. Sci. (Phys.-Math.), Institute of Mathematics, Ufa Federal Research Centre, RAS, 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 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.