Unitary subgroups and orbits of compact self-adjoint operators
Volume 238 / 2017
Abstract
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.
