On the Bochner theorem on orthogonal operators

Abstract:

We prove the following theorem. Any isometric operator $U$, that acts from the Hilbert space $H_1(\Omega )$ with nonnegative weight $p(x)$ to the Hilbert space $H_2(\Omega )$ with nonnegative weight $q(x)$, allows for the integral representation \[ Uf=\frac {1}{q(\xi )} \frac {\partial ^n}{\partial \xi _1\ldots \partial \xi _n}\int _{\Omega } \overline {L(\xi ,t)}f(t)p(t)dt, \] \[ U^{-1}f= \frac {1}{p(\xi )}\frac {\partial ^n}{\partial \xi _1\ldots \partial \xi _n} \int _{\Omega }\overline {K(\xi ,t)}f(t)q(t)dt, \] where the kernels $L(\xi ,t)$ and $K(\xi ,t)$ satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.