Applications of the Projection Characterization and Symmetry Bootstrap for Operators that are Nearby Commuting Operators

This paper applies the Projection Characterization and the Symmetry Bootstrap for two operators $A,B$ (with $A$ unitary or self-adjoint) nearby commuting operators $A^{\mathrm{\prime }},B^{\mathrm{\prime }}$ (with $A^{\mathrm{\prime }}$ unitary or self-adjoint) from arXiv:2412.20795 to problems related to structured almost commuting unitaries, self-adjoint operators that almost anti-commute, and unitaries that almost commute up to a rational phase. We prove several new stability theorems for these types of relations for matrices. We also provide estimates for how close the constructed operators are for several known stability theorems that did not previously have asymptotic estimates.