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

TL;DR

This paper extends the diagonalization of compact operators from commutative to finite non-commutative W*-algebras within Hilbert modules, providing a framework for analyzing operators like the Schrödinger operator in magnetic fields.

Contribution

It generalizes the diagonalization of compact operators to Hilbert modules over finite W*-algebras, including non-commutative cases, and explores applications to magnetic Schrödinger operators.

Findings

Diagonalization of compact operators in non-commutative W*-algebra modules

Existence of eigenvectors and eigenvalues in Hilbert modules

Application to magnetic Schrödinger operators in irrational rotation algebra

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.