On the Extension of a Certain Class of Carleman Operators

Abstract

The integral operators of Carleman play an important role in the spectral theory of selfadjoint operators and made the object of several works such those of G. I. Targonski [Proc. Amer. Math. Soc. 18 (1967)(3)], V. B. Korotkov [Sib. Math. J. 11 (1970)(1)], and J. Weidmann [Manuscripta Math. (1970)(2)]. In the present paper, we study a certain class of these operators in the Hilbert space $L^2\left(X,\mu \right)$. Precisely, we give necessary and sufficient conditions so that they possess equal defeciency indices. Such operators find their applications in the theory of random variable approximation.