Unitary Group Orbits Versus Groupoid Orbits of Normal Operators

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