A class of a non-Banach F-lattices E such that every orthomorphism on E is central with applications to Musielak–Orlicz sequence spaces

Abstract

Let E=(E,‖⋅‖) be a non-Banach F-lattice non-containing a subspace linearly homeomorphic to the F-lattice ω:=RN. The symbol Orth(E) denotes the linear lattice of all orthomorphisms on E, and Z(E) denotes the principal ideal of Orth(E) generated by the identity IE on E. In this paper, we give a sufficient condition for the equality Orth(E)=Z(E) to hold. In particular, this identity holds true for every symmetric sequence space E. We also obtain a useful series-criterion for the class of discrete F-lattices E with order continuous norm for the identity Orth(E)=Z(E) to hold. These results apply non-trivially for the class of Musielak-Orlicz sequence spaces. We give a few examples illustrating our main results. At the end of this paper, we set a few open questions about orthomorphisms.