Unitary group orbits versus groupoid orbits of normal operators

Abstract

We study the unitary orbit of a normal operator $a\in \mathcal B(\mathcal H)$, regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the differing ambient spaces if and only if the spectrum of $a$ is finite, and in that case it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of $a$ is not finite, we describe the closure of the orbits of $a$ for the different norm topologies involved. We relate these results to the action of the groupoid of the partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator $a$ we also describe the norm closure of its groupoid orbit ${\mathcal O}_a$, which leads to necessary and sufficient spectral conditions on $a$ ensuring that ${\mathcal O}_a$ is norm closed and that ${\mathcal O}_a$ is a closed embedded submanifold of $\mathcal B(\mathcal H)$.