On the coincidence of reproducing kernel Hilbert spaces connected by a special transformation

V. V. Napalkov ; V. V. Napalkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 149-159

Résumé

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 ${\mathcal A}$ 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 $H_2$.

Détail

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