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Local operators and a characterization of the Volterra operator. (English) Zbl 1216.47091

The article deals with operators D m K, where D is the differentiation operator and K:C 0 ([a,b])C m ([a,b]). The main result is the following: if D m K is locally defined, then there exists a continuous function h:[a,b]× such that, for all ϕC m [[a,b]),

K(ϕ)(x)=1 (m-1)! 0 x (x-t) m h(t,ϕ(t))dt+ k=0 m-1 (D k K)(ϕ)(a) k!(x-a) k ·

As a corollary, a characterization of the Volterra operator K:C 0 ([a,b])C 1 ([a,b]) is obtained. In the end of the article, it is mentioned that any operator K mapping the set of all real analytic functions defined on [a,b] in the set of all real functions defined on [a,b] is locally defined.

47H30Particular nonlinear operators
47A67Representation theory of linear operators