On products of symmetries acting on Hilbert spaces

TL;DR

The paper investigates when unitary operators on a Hilbert space can be expressed as products of a small number of symmetries, focusing on the elusive ${\rm Sym}_3$-class. It develops a standard form for symmetries relative to space decompositions, derives spectral constraints for ${\rm Sym}_3(\mathbb C^d)$-elements with two eigenvalues, and introduces a purity condition (Condition (P)) to guide construction. It then provides general obstructions and positive results for when unitary orbits coincide with ${\rm Sym}_k$-orbits, including a sharp characterization for Sym_1: ${\rm Sym}_1(T)=\mathcal U(T)$ if and only if $T$ is normal with at most two eigenvalues, and several cases where ${\rm Sym}_2(T)=\mathcal U(T)$ or ${\rm Sym}_3(\mathbb C^d)$ includes a broad class of unitaries. The work connects operator algebra techniques, spectral theory, and the geometry of unitary orbits, offering both concrete criteria and open problems for higher k and more general algebras.

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 $k \le 3$.