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Unitary subgroups and orbits of compact self-adjoint operators

Tom 238 / 2017

Tamara Bottazzi, Alejandro Varela Studia Mathematica 238 (2017), 155-176 MSC: Primary 22F30, 22E10, 51F25; Secondary 47B15. DOI: 10.4064/sm8690-12-2016 Opublikowany online: 21 April 2017

Streszczenie

Let $\mathcal {H}$ be a separable Hilbert space, and let $\mathcal {D}(\mathcal {B}(\mathcal {H})^{ah})$ be the anti-Hermitian bounded diagonal operators in some fixed orthonormal basis and $\mathcal {K}(\mathcal {H})$ the compact operators. We study the group of unitary operators $$ {\mathcal {U}}_{k,d}=\{u\in \mathcal {U}(\mathcal {H}): \exists D\in \mathcal {D}(\mathcal {B}(\mathcal {H})^{ah}),\, u-e^D \in \mathcal {K}(\mathcal {H})\} $$ in order to obtain a concrete description of short curves in unitary Fredholm orbits $\mathcal {O}_b=\{ e^K b e^{-K}:K\in \mathcal {K}(\mathcal {H})^{ah}\}$ of a compact self-adjoint operator $b$ with spectral multiplicity one. We consider the rectifiable distance on $\mathcal {O}_b$ defined as the infimum of curve lengths measured with the Finsler metric defined by means of the quotient space $\mathcal {K}(\mathcal {H})^{ah}/\mathcal {D}(\mathcal {K}(\mathcal {H})^{ah})$. Then for every $c\in \mathcal {O}_b$ and $x\in T_c(\mathcal {O}_b) $ there exists a minimal lifting $Z_0\in \mathcal {B}(\mathcal {H})^{ah}$ (in the quotient norm, not necessarily compact) such that $\gamma (t)=e^{t Z_0} c e^{-t Z_0}$ is a short curve on $\mathcal {O}_b$ in a certain interval.

Autorzy

  • Tamara BottazziInstituto Argentino de Matemática
    “Alberto P. Calderón”
    Saavedra 15 3$^{\rm er}$ piso
    (C1083ACA) Buenos Aires, Argentina
    e-mail
  • Alejandro VarelaInstituto de Ciencias
    Universidad Nacional de General Sarmiento
    J. M. Gutierrez 1150
    (B1613GSX) Los Polvorines
    Pcia. de Buenos Aires, Argentina
    and
    Instituto Argentino de Matemática
    “Alberto P. Calderón”
    Saavedra 15 3$^{\rm er}$ piso
    (C1083ACA) Buenos Aires, Argentina
    e-mail

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