Different aspects of stochastic observability of dynamical systems described by semigroups of linear operators are discussed. First, basic notions taken from the theory of linear operators and semigroups are presented. Next, using methods of functional analysis, necessary and sufficient conditions for stochastic observability are formulated and proved. The problem of minimization of the number of measurements needed for stochastic observability is discussed with the aid of linear algebra methods. The minimal number of moments of observations are estimated. Finally, observability of Chebyshev systems is considered and an effevtive method for its investigation is proposed. Observability of systems with the so-called Hessenberg generator is also considered.
In addition the paper contains several remarks and comments concerning stochastic observability, and its connections with stochastic controllability are also presented. The relationships to the results on stochastic observability given in the literature are mentioned. The stochastic dynamical systems considered in the paper can also be found in the monograph [R. S. Ingarden, A. Kossakowski and M. Ohya, Information dynamics and open systems (Kluwer Academic Publishers, Dordrecht) (1997; Zbl 0891.94007)].