REPRESENTATION OF ATOMIC OPERATORS AND EXTENSION PROBLEMS

The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiiˇ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some $\sigma$-algebra to a larger space of functions measurable with respect to a larger $\sigma$-algebra, as well as to the possibility of extending any $\sigma$-homomorphism from a smaller-measure algebra to a $\sigma$-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective $\sigma$-algebras which provides such possibilities and induces the positive answer to the above representation problem.AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05

MEAN VALUES OF FIELD AND ATOMIC OPERATOR PRODUCTS IN THE PROBLEMS OF QUANTUM OPTICS

Exact expressions for the mean values of the products of field and atomic operators are obtained in the general case of a chaotic, coherent or squeezed initial state of the field. These expressions permit one to eliminate the field variables from the dynamical equations.