Opuscula Math. 30, no. 3 (2010), 241-248
http://dx.doi.org/10.7494/OpMath.2010.30.3.241

Opuscula Mathematica

Uniformly continuous set-valued composition operators in the space of total φ-bidimensional variation in the sense of Riesz

Wadie Aziz
José Giménez
Nelson Merentes
José Luis Sánchez

Abstract. In this paper we prove that if a Nemytskij composition operator, generated by a function of three variables in which the third variable is a function one, maps a suitable large subset of the space of functions of bounded total $$\varphi$$-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting.

Keywords: $$\varphi$$-bidimensional variation, uniformly continuous, Nemytskij operator.

Mathematics Subject Classification: 47H30, 39B52.

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• Wadie Aziz
• Universidad de Los Andes, Dpto. de Física y Matemáticas, Trujillo - Venezuela
• José Giménez
• Universidad de Los Andes, Facultad de Ciencias, Dpto. de Matemáticas, Mérida - Venezuela
• Nelson Merentes
• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas - Venezuela
• José Luis Sánchez
• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas - Venezuela
• Received: 2010-01-12.
• Revised: 2010-04-03.
• Accepted: 2010-04-06.

Cite this article as:
Wadie Aziz, José Giménez, Nelson Merentes, José Luis Sánchez, Uniformly continuous set-valued composition operators in the space of total φ-bidimensional variation in the sense of Riesz, Opuscula Math. 30, no. 3 (2010), 241-248, http://dx.doi.org/10.7494/OpMath.2010.30.3.241

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