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Zbl 1216.47091
Matkowski, J.
Local operators and a characterization of the Volterra operator.
(English)
[J] Ann. Funct. Anal. AFA 1, No. 1, 36-40, electronic only (2010). ISSN 2008-8752/e

The article deals with operators $D^m \circ K$, where $D$ is the differentiation operator and $K: C^0([a,b]) \to C^m([a,b])$. The main result is the following: if $D^m \circ K$ is locally defined, then there exists a continuous function $h: [a,b] \times {\Bbb R} \to {\Bbb R}$ such that, for all $\varphi \in C^m[[a,b])$, $$K(\varphi)(x) = \frac1{(m-1)!} \int_0^x (x - t)^m h(t,\varphi(t)) \, dt + \sum_{k=0}^{m-1} \frac{(D^k \circ K)(\varphi)(a)}{k!} (x - a)^k.$$ As a corollary, a characterization of the Volterra operator $K: C^0([a,b]) \to C^1([a,b])$ 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.
[Peter Zabreiko (Minsk)]
MSC 2000:
*47H30 Particular nonlinear operators
47A67 Representation theory of linear operators

Keywords: Nemytskij operator; locally defined operator; superposition operator; Volterra operator; differentiable functions; analytic functions

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