On products of symmetries acting on Hilbert spaces

Abstract

Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with $J^2=I$. Denote by ${\text{Sym}}_k(\mathcal{H})$ the subset of $\mathcal{U}(\mathcal{H})$ consisting of those operators expressible as a product of $k$ symmetries. It is known that $\mathcal{U}(\mathcal{H})={\text{Sym}}_4(\mathcal{H})$ if $\dim \mathcal{H}=\infty$, while the only additional condition in finite dimensions is that the determinant be $\pm 1$. Of all the sets ${\text{Sym}}_k(\mathcal{H})$ with $k\in \{1,2,3,4\}$, the case $k=3$ has been the most stubborn to characterise. Among other things, we investigate which elements of ${\text{Sym}}_3(\mathcal{H})$ possess exactly two eigenvalues in the setting where $\mathcal{H}$ is finite-dimensional. We also consider the problem: when is the unitary orbit of an operator $T$, i.e., the set \[ \{ U^* T U : U \in \mathcal{U}(\mathcal{H}) \} \] the same as its ${\text{Sym}}_k$-orbit, i.e., the set \[ \{ U^* T U: U \in \text{Sym}_k(\mathcal{H})\} ? \] Clearly, the cases of interest are when $k\le 3$.