Linear Volterra operators in some metric spaces

Abstract

We apply our definition of Volterra operator on abstract spaces to some problems arising in metric spaces. In contrast to those known before, our definition requires only the existence of a σ-algebra on a metric space. Note that, being applied to such spaces, the new definition substantially extends the classes of operators of an evolutionary nature. It also allows one to relate different properties of the Volterra-type operators. In particular, the problem of quasi-nilpotentness studied traditionally in the Banach spaces only (since it requires equality to zero of the spectral radius of an operator) allows interpretation in complete locally convex spaces. Apparently, the question on preserving the Volterra property by a conjugate operator is posed for the first time. It should be mentioned that, generally speaking, the dual space to a Frechet (i.e., complete locally convex) space is not a Frechet space.