2003/04/07 by Ronald G. Douglas, Douglas, Ronald G., Gadadhar Misra +1
Mathematics · #46 E22 #47 A13 #47 A20 #Advanced Operator Algebra Research #Analytic and geometric function theory #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory (math.SP) #math.SP #msc:46 #msc:47 #msc:A13 #msc:A20 #msc:E22
paper · pdf · doi:10.48550/arxiv.math/0304084
arxiv created 2003/04/07 · openalex publication_date 2003/04/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of a quasi-free Hilbert module over a function algebra A consisting of holomorphic functions on a bounded domain Ω in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra A. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module M is said to be compactly supported if there exists a constant β satisfying ‖ϕf ‖ ≤ β‖ϕ‖X ‖f‖ for some compact subset X of Ω and ϕ in A, f in M. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.