article · Taiwanese Journal of Mathematics · 2011
Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces
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Abstract
Let $X, Y$ be linear topological subspaces of the space of all measurable functions over a $\sigma$-finite, atomless and complete measure space $\Omega$. The question of existence of a nonzero, linear continuous operator $T: X \to Y$ is a natural extension of the question "does X admit a nonzero, linear continuous functional?". In case of $X = L^\Phi(\Omega)$, $Y = L^\Psi(\Omega)$ being Orlicz spaces Ph.Turpin ([11]) gave a criterion telling when there is no nonzero, linear and continuous operator between $L^\Phi(\Omega)$ and $L^\Psi(\Omega)$. That result was extended to the case of Musielak-Orlicz spaces by A.K.Kalindé, R.Płuciennik and M.Wisła ([2, 8]) but only necessary conditions have been presented by them - in fact conditions assuring that there is no nonzero, linear and continuous operator between $L^\Phi(\Omega)$ and $L^\Psi(\Omega)$. In this paper we give inverse theorems to the Kalindé and Płuciennik-Wisła theorems. Generally speaking we state that there exists a nonzero, linear and continuous operator if and only if it there exists a set $A \subset \Omega$ of positive and finite measure such that the inclusion operator $i: L^\Phi(A) \to L^\Psi(A)$ is nonzero and continuous.
Cite this paper
Kasior, E., & Wisła, M. (2011). Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces. Taiwanese Journal of Mathematics, 15(3). https://doi.org/10.11650/twjm/1500406305
Kasior, Ewa, and Marek Wisła. “Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces.” Taiwanese Journal of Mathematics, vol. 15, no. 3, 2011, https://doi.org/10.11650/twjm/1500406305.
Kasior, Ewa, and Marek Wisła. 2011. “Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces.” Taiwanese Journal of Mathematics 15 (3). https://doi.org/10.11650/twjm/1500406305.
Kasior, E. and Wisła, M. (2011) ‘Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces’, Taiwanese Journal of Mathematics, 15(3). Available at: https://doi.org/10.11650/twjm/1500406305.
@article{kasior2011existence,
title = {{Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces}},
author = {Kasior, Ewa and Wisła, Marek},
journal = {Taiwanese Journal of Mathematics},
volume = {15},
number = {3},
year = {2011},
doi = {10.11650/twjm/1500406305}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.