Perturbation of orthonormal bases with an application to diagonalization

Given an orthonormal basis {e n } n=1 (infinity) in a Hilbert spaceH, and a dense linear manifold D (subset of) H, we show that there exists a unitary operator V on H such thatI-V is a trace-class operator with arbitrarily small trace norm, and Ve j (element of) D for allj. This result can be used to simplify certain arguments of J. Xia concerning the simultaneous diagonalization of operators on a space of square integrable functions.