Boundedness and continuity of superposition operator on Er(p) and Fr(p)

Abstract

Let X ∈ {Er(p)V , Fr(p)}, in this research, necessary and sufficient conditions are given for superposition operator to act from X into the space l1. Moreover, necessary and sufficient conditions are obtained for superposition operator acting from X into l1 to be locally bounded, bounded, and continuous. Suppose that Pf is a superposition operator which acts from X into l1 , it is found that1. Pf is locally bounded if and only if f satisfies the condition A(2 / ) ,2. if Pf is bounded then f satisfies the condition A(2 / ) ,3. Pf is continuous if and only if f satisfies the condition A(2) .

Keywords

• sequence space
• superposition operator
• locally bounded function
• bounded function
• continuous function