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Unitary Group Orbits Versus Groupoid Orbits of Normal Operators

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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 various 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 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 \({{\mathcal {B}}}({{\mathcal {H}}})\).

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Acknowledgements

The research of the first-named author (D.B.) was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0878, within PNCDI III. The research of the second-named author (G.L.) was supported by CONICET, ANPCyT, and Universidad de Buenos Aires (Argentina). The authors would like to thank both referees for their helpful corrections and comments, which improved the quality of the manuscript.

Funding

Funding was provided by Secretaria de Ciencia y Tecnica, Universidad de Buenos Aires (Grant no. PICT 2019-04060, Consejo Nacional de Investigaciones Cientficas y Tcnicas, AR (Grant no. PIP 2016-112201, Ministry of Research, Innovation and Digitization, RO (Grant no. PN-III-P4-ID-PCE-2020-0878)).

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Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, Bucharest, Romania

    Daniel Beltiţă

  2. Departamento de Matemática, FCEyN, Universidad de Buenos Aires (UBA), Buenos Aires, Argentina

    Gabriel Larotonda

  3. Instituto Argentino de Matemática, CONICET, Buenos Aires, Argentina

    Gabriel Larotonda

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  1. Daniel Beltiţă
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Correspondence to Gabriel Larotonda.

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Beltiţă, D., Larotonda, G. Unitary Group Orbits Versus Groupoid Orbits of Normal Operators. J Geom Anal 33, 95 (2023). https://doi.org/10.1007/s12220-022-01187-5

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