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Annals of Global Analysis and Geometry

, Volume 13, Issue 3, pp 207–226 | Cite as

Diagonalization of compact operators in Hilbert modules over finiteW*-algebras

  • V. M. Manuilov
Article

Abstract

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH*A dual toHA under slight restrictions one can find a set of “eigenvectors”xi εH*A and a non-increasing sequence of “eigenvalues” λ i εA such thatK xi=xi λ i and the selfdual HilbertA-module generated by these “eigenvectors” is the wholeH*A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA θ and discuss the possibility of its diagonalization.

Key words

Diagnoalization of operators Hilbert module compact operator W*-algebras 

MSC 1991

46L89 

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • V. M. Manuilov
    • 1
  1. 1.Moscow State Public UniversityMoscowRussia

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