2025/11/15 by Laurent W. Marcoux, Heydar Radjavi, Marcoux, Laurent W. +3
Mathematics · #Spectral Theory in Mathematical Physics #Advanced Operator Algebra Research #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2511.12028
Let H be a complex, separable Hilbert space (of finite or infinite dimension), and let U(H) denote the group of unitary operators on H. A symmetry is, by definition, a unitary operator J with J2 =I. Denote by Symk(H) the subset of U(H) consisting of those operators expressible as a product of k symmetries. It is known that U(H) = Sym4(H) if dim H = ∞, while the only additional condition in finite dimensions is that the determinant be ± 1. Of all the sets Symk(H) with k ∈ \ 1, 2, 3, 4\, the case k =3 has been the most stubborn to characterise. Among other things, we investigate which elements of Sym3(H) possess exactly two eigenvalues in the setting where H is finite-dimensional. We also consider the problem: when is the unitary orbit of an operator T, i.e., the set \ U^* T U : U ∈ U(H) \ the same as its Symk-orbit, i.e., the set \ U^* T U: U ∈ Symk(H)\ ? Clearly, the cases of interest are when k ≤ 3.