Approximate versions of triangularizing or reducing conditions (H. Radjavi)

In the past few years work has been done by several authors (including Bernik, Drnovšek, Kokol-Bukovšek, Košir, Marcoux, Mastnak, Omladič, Radjavi) in demonstrating that certain approximate versions of hypotheses in previously known results can replace the exact versions. To illustrate with a question: what can be said about a group or semigroup S of matrices if, for all A and B in S, the commutator AB -BA is "small" in some sense? Bernik and Radjavi have dealt with this question when smallness is measured in terms of norm or spectral radius; Mastnak and Radjavi have studied it when smallness means that the spectrum is concentrated on one line.