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UNIFORMLY BOUNDED COMPOSITION OPERATORS

Part of: Nonlinear operators and their properties Commutative Banach algebras and commutative topological algebras Real functions Special classes of linear operators

Published online by Cambridge University Press:  30 July 2015

DOROTA GŁAZOWSKA  and
JANUSZ MATKOWSKI
DOROTA GŁAZOWSKA*
Affiliation:
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland email [email protected]
JANUSZ MATKOWSKI
Affiliation:
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland email [email protected]
*
✉[email protected]
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Abstract

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We prove that if a uniformly bounded (or equidistantly uniformly bounded) Nemytskij operator maps the space of functions of bounded ${\it\varphi}$-variation with weight function in the sense of Riesz into another space of that type (with the same weight function) and its generator is continuous with respect to the second variable, then this generator is affine in the function variable (traditionally, in the second variable).

Keywords

composition operatorNemytskij operatorbounded operatoruniformly bounded operator𝜙-variation in the sense of Rieszweight function

MSC classification

Primary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
Secondary: 26A45: Functions of bounded variation, generalizations 46J10: Banach algebras of continuous functions, function algebras 47B38: Operators on function spaces (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 463 - 469
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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