Diagonalization of compact operators in Hilbert modules over finite W*-algebras

Abstract

It is known that a continuous family of compact operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators in Hilbert modules over a commutative W*-algebra. The aim of the present paper is to generalize this fact for a finite W*-algebra $A$ not necessarily commutative. We prove that for a compact operator $K$ acting in the right Hilbert $A$-module $H^*_A$ dual to $H_A$ under slight restrictions one can find a set of "eigenvectors" $x_i\in H^*_A$ and a non-increasing sequence of "eigenvalues" $\lambda_i\in A$ such that $K\,x_i = x_i\,\lambda_i$ and the autodual Hilbert $A$-module generated by these "eigenvectors" is the whole $H_A^*$. As an application we consider the Schr\"odinger operator in magnetic field with irrational magnetic flow as an operator acting in a Hilbert module over the irrational rotation algebra $A_{\theta}$ and discuss the possibility of its diagonalization.