Diagonalization modulo norm ideals with Lipschitz estimates (Q1355078)

Motivated by earlier results of Weyl, von Neumann, Berg and Voiculescu, the author investigates simultaneous diagonalization of multiplication operators with respect to a given ideal of operators. He replaces tuples of commuting operators considered by Voiculescu by multiplication operators coming from Lipschitz functions with constant at most one. He investigates a functional calculus for this class yielding for each function a diagonal operator built up from certain values of the given function. The multiplication operators are considered on the \(L^2\)-spaces with respect to a fixed measure on the underlying compact Hausdorff space. The author characterizes those measures which allow a diagonalization with respect to a fixed ideal norm in terms of `good' approximate units. He shows that the class of Lipschitz functions can be diagonalized modulo operators in the Schatten \(p\)-class for every measure if the \(p\)-dimensional Hausdorff measure is finite. In particular, this applies for an \(n\)-tuple of commuting operators for \(p=n\). For a fixed measure, he can prove simultaneous diagonalization with respect to the Schatten \(p\)-class if the volume of balls with radius \(r\) is not essentially smaller than \(r^p\). He also investigates diagonalization with respect to the Schatten-Lorentz spaces, where the second index is either one or infinity and gives examples where no simultaneous diagonalization is possible for the Schatten-Lorentz class \(S_{2,1}\).