Characterization of Carleman Operators in Riesz Spaces

Abstract

This chapter discusses the characterization of Carleman operators in Riesz spaces. It describes the way the general theory of Riesz spaces (and in particular Banach lattices) can be used as an approach to study a classical type integral operator and its generalization. The condition for a linear operator T : L 2 ( Y , ν ) → L 2 ( X , μ ) for being a Carleman integral operator is described in the chapter. The investigation into the properties of this type of operator dates back to the work of Carleman, Stone, and von Neumann. In the results mentioned in the chapter for this investigation, the presence of a measure space is essential for the definition of a Carleman (integral) operator. The investigation is taken further by defining the concept of a Carleman operator from a normed space into an Archimedean Riesz space by taking the Korotkov condition as the basis of given definition. The chapter also discusses the conditions for a subset H ⊂ E of a Dedekind complete Riesz space E to be equimeasurable.