Unitary subgroups and orbits of compact self-adjoint operators

Tamara Bottazzi ¹   ; Alejandro Varela ²

¹ Instituto Argentino de Matemática “Alberto P. Calderón” Saavedra 15 3$^{\rm er}$ piso (C1083ACA) Buenos Aires, Argentina
² Instituto 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

Studia Mathematica, Tome 238 (2017) no. 2, pp. 155-176

Résumé

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.

Détail

DOI : 10.4064/sm8690-12-2016

Keywords: mathcal separable hilbert space mathcal mathcal mathcal anti hermitian bounded diagonal operators fixed orthonormal basis mathcal mathcal compact operators study group unitary operators mathcal mathcal mathcal exists mathcal mathcal mathcal u e mathcal mathcal order obtain concrete description short curves unitary fredholm orbits mathcal e k mathcal mathcal compact self adjoint operator spectral multiplicity consider rectifiable distance mathcal defined infimum curve lengths measured finsler metric defined means quotient space mathcal mathcal mathcal mathcal mathcal every mathcal mathcal there exists minimal lifting mathcal mathcal quotient norm necessarily compact gamma t short curve mathcal certain interval