2005/07/26 by Ronald G. Douglas, Douglas, Ronald G. · 1 citation
Mathematics · #47B99 #47L15 #47L80 #58J20 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.FA #math.OA #msc:47B99 #msc:47L15 #msc:47L80 #msc:58J20
paper · pdf · doi:10.48550/arxiv.math/0507542
Twelve pages
arxiv created 2005/07/26 · openalex publication_date 2005/07/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator theoretic results. In [11] Berger and Shaw established that finitely cyclic hyponormal operators have trace-class self-commutators. In [9], [31] Berger and Voiculescu extended this result to operators whose self-commutators can be expressed as the sum of a positive and a trace-class operator. In this note we show this result can't be extended to operators whose self-commutator can be expressed as the sum of a positive and a Sp-class operator. Then we discuss a conjecture of Arveson [4] on homogeneous submodules of the m-shift Hilbert space H2m and propose some refinements of it. Further, we show how a positive solution would enable one to define K-homology elements for subvarieties in a strongly pseudo-convex domain with smooth boundary using submodules of the corresponding Bergman module. Finally, we discuss how the Chern character of these classes in cyclic cohomology could be defined and indicate what we believe the index to be.