Integral Equations and Operator Theory

, Volume 47, Issue 4, pp 435–456

Quasi-Free Resolutions Of Hilbert Modules

Original paper

DOI: 10.1007/s00020-003-1169-4

Cite this article as:
Douglas, R. & Misra, G. Integr. equ. oper. theory (2003) 47: 435. doi:10.1007/s00020-003-1169-4

Abstract.

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ 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 $\mathcal{A}$. 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 $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. 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.

Mathematics Subject Classification (2000):

Primary: 47A1347A20Secondary: 46E22

Keywords:

Hilbert ModuleKernel functionHomological methodsQuasi-free modulesCompactly supported modules

Copyright information

© Birkhäuser-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Indian Statistical InstituteBangaloreIndia