Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces

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.