Some remarks on substitution and composition operators. (English) Zbl 1521.47045
Rend. Ist. Mat. Univ. Trieste 53, Paper No. 6, 25 p. (2021); corrigendum ibid. 53, Paper No. 8, 2 p. (2021).
In this paper, the authors study the linear substitution operator and nonlinear composition operator, namely,
\[
S_\varphi(f)=f\circ\varphi,\; \varphi:[0,1]\rightarrow [0,1],
\]
and
\[
C_g(f)=g\circ f,\; g:\mathbb{R}\rightarrow \mathbb{R},
\]
respectively. They show that these operators have a very different behavior in the space of continuous functions, Lipschitz functions, functions of bounded variation, and Baire class one functions. Moreover, they give examples and counterexamples which illustrate this behavior.
Reviewer: Mohamed Abdalla Darwish (Damanhour)
MSC:
| 47B33 | Linear composition operators |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 26A16 | Lipschitz (Hölder) classes |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
| 26A45 | Functions of bounded variation, generalizations |
Keywords:
substitution operators; composition operators; injectivity; surjectivity; continuous functions; Lipschitz functions; functions of bounded variation; Baire one functionsReferences:
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