Two‐Sided Closed Ideals of Certain Algebras of Unbounded Operators

Abstract

Abstract This paper investigates the two‐sided uniformly closed ideals of the maximal Op*‐algebra L + ( D ) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non‐trivial two‐sided closed ideals of L + ( D ) is well‐ordered by inclusion and the α‐th closed ideal 𝔍 α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then 𝔍 α . An element A in L + ( D ) belongs to the minimal closed ideal 𝔍 0 if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.