Perturbation of orthonormal bases with an application to diagonalization. (Q1865910)

Given an orthonormal basis \(\{e_n\}_{n=1}^\infty\) in a Hilbert space \(H\) and a dense linear manifold \(D\subset H\), the authors show that there exists a unitary operator \(V\) on \(H\) such that \(I-V\) is a trace class operator with arbitrarily small trace norm, and \(Ve_j\in D\) for all \(j\). This result can be used to simplify certain arguments of \textit{J.-B. Xia} [J. Funct. Anal. 145, 491--526 (1997; Zbl 0880.47027)] concerning the simultaneous diagonalization of operators on a space of square integrable functions.