Diagonalization and unitary equivalence modulo Schatten \(p\)-classes (Q1584562)

The autor gives a necessary and sufficient criterion for a commuting couple \(A= (A_1,\dots, A_n)\) of selfadjoint operators on a Hilbert space \(H\) to be simultaneously diagonalizable modulo the Schatten-class \({\mathcal C}_p\) provided that \(1<p<\infty\). The condition is that \[ \int^1_0 [\mu_\xi(B(x, r))/r^p]^{1/(p- 1)}dr/r< \infty \] holds for every nonzero \(\xi\in H\), \(\mu_\xi\) a.e. in \(x\in H\). Here \(B(x,r)\) is the ball of radius \(r\) centered at \(x\) and the measure \(\mu_\xi\) is given via the spectral resolution \({\mathcal E}\) of \(A\) by \(\mu_\xi(E)= \langle{\mathcal E}(E)\xi, \xi\rangle\). This paper is in the line of Voiculescu's work on simultaneous diagonalization of commuting operators.