Even positive-definite hyperbolically convex functions in a Hilbert space. Representations of Hilbert spaces by self-adjoint operators

Abstract

UDC 517.98

The paper consists of two parts. In the first part, we establish sufficient and necessary conditions for the integral representation of even positive-definite hyperbolically convex (h.c.) functions $k(x),$ $x\in H$.  These functions are continuous in the $j$-topology.  The positive definiteness of a function is understood in the pointwise sense.  The analyzed theorem is a modified version of the Berezansky theorem presented  in [Yu. M. Berezansky and I. M. Gali, Ukr. Mat. Zh., 24, № 4, 351–372 (1972)]. The integral representation for some other positive-definite kernels were considered in  [O. V. Lopotko and I. I. Rudynskyi, Ukr. Mat. Zh., 34, № 3, 310–312 (1982)] and [O. V. Lopotko, Dop. Akad. Nauk Ukr., Ser. A, 8, 11–13 (1991)].

In the second part, we prove the integral representation for a family of commutative self-adjoint operators $\mathcal{A}_x$ connected by algebraic relations. For this purpose, we construct a rigging (chain) $H_{\kappa}=H_0\supset H_+=L_2\supset \mathcal{D}$ for $x\in H.$ Our proof is based on the integral representation of even positive-definite h.c. functions of infinitely many variables (see [Yu. M. Berezansky and Yu. G. Kondratyev, Spectral methods in infinite-dimensional analysis, Naukova Dumka, Kyiv (1988)] and  [O. V. Lopotko, Bukov. Mat. Zh., 11, № 1, 26–38 (2023)]).  Some other forms of generalizations of this kind were considered in  [Yu. M. Berezansky and A. A. Kalyuzhny, Ukr. Mat. Zh., 36, № 4, 417–421 (1984);  A. A. Kurepa, Canad. Math. J., 12, 45–50 (1960)]; and [Yu. S. Samoilenko, Spectral Theory of the Sets of Self-Adjoint Operators, Naukova Dumka, Kyiv (1984)].