- Lebesgue decomposition and Gleason — Whitney property of operator algebra (arXiv)
Author: Raphael Cruator, Michael Hertz
Abstract: Broadly speaking, this paper is about the dual space of operator algebras. More precisely, we investigate the existence of what are called Lebesgue projections, i.e., central projections in the duality of operator algebras that detect weakly continuous parts of the dual space. Associated with such a projection is a Lebesgue decomposition of the dual space. We are particularly interested in Lebesgue projections in the context of the inclusion of operator algebras. We show how their existence is closely related to the extended property of inclusion, which is reminiscent of the classic theorem of Gleason and Whitney. We show that this Gleason-Whitney property fails in many examples of concrete operator algebras of functions. This partially explains why there is a lack of compatible Lebesgue decompositions and highlights that the classical inclusion H∞⊂L∞ on circles does not exhibit general behavior. △
2. Lebesgue-type decomposition of linear relations and Ando's uniqueness criterion (arXiv)
Author: Seppo Hassi, Zoltan Sebestien, Henk de Snu
Summary: : A linear relation, i.e. a multivalued operator T from a Hilbert space H to a Hilbert space K, has a Lebesgue-type decomposition T=T1+T2. Here, T1 is a closable operator and T2 is a singular operator or relation. There is one standard decomposition of T, called the Lebesgue decomposition, whose closable parts are characterized by maximality among all closable parts in the sense of dominance. All Lebesgue-type decompositions are parameterized, which also provides necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are obtained with a weaker Lebesgue-type decomposition. Here, T1 is not necessarily closable, it is just an operator. Additionally, occlusion can be characterized in a variety of useful ways. In the special case of range-space relations, the above decomposition is applicable when dealing with pairs of (non-negative) bounded operators and non-negative forms, and when dealing with classical frameworks of positive scale. Masu.
