# Basic properties of multiplication and composition operators between distinct Orlicz spaces

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Revista Matemática Complutense

, Volume 30, Issue 2, pp 335–367

First Online: 15 December 2016Received: 31 March 2016Accepted: 16 November 2016DOI: 10.1007-s13163-016-0214-1

Cite this article as: Chawziuk, T., Estaremi, Y., Hudzik, H. et al. Rev Mat Complut 2017 30: 335. doi:10.1007-s13163-016-0214-1

Abstract

First, we present some simple and easily verifiable necessary conditions and sufficient conditions for boundedness of the multiplication operator \M u\ and composition operator \C T\ acting from Orlicz space \L^{\Phi 1}\Omega \ into Orlicz space \L^{\Phi 2}\Omega \ over arbitrary complete, \\sigma \-finite measure space \\Omega ,\Sigma ,\mu \. Next, we investigate the problem of conditions on the generating Young functions, the function u, and-or the function \h=d\mu \circ T^{-1}-d\mu \, under which the operators \M u\ and \C T\ are of closed range or finite rank. Finally, we give necessary and sufficient conditions for boundedness of the operators \M u\ and \C T\ in terms of techniques developed within the theory of Musielak–Orlicz spaces.

KeywordsMultiplication operator Composition operator Continuous operators Closed-range operators Finite-rank operators Orlicz space Musielak–Orlicz space Lebesgue space Measurable transformation Radon–Nikodym derivative Mathematics Subject Classification47B38 47B33 46E30

Autor: T. Chawziuk - Y. Estaremi - H. Hudzik - S. Maghsoudi - I. Rahmani