In the present paper, we study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group ${\mathcal{U}}_{\mathcal{K}+\mathbb{C}}$ of the unitization of the compact operators $\mathcal{K}(\mathcal{H})+\mathbb{C}$, or equivalently, the quotient ${\mathcal{U}}_{\mathcal{K}+\mathbb{C}}/{\mathcal{U}}_{\mathcal{D}(\mathcal{K}+\mathbb{C})}$. We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of $\mathcal{K}(\mathcal{H})+\mathbb{C}$.

中文翻译：

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的自伴随算子的酉轨道中的测地线邻域 $\mathcal{钾}+\mathbb{C}$

在本文中，我们研究了在酉群作用下具有谱重数为 1 的紧 Hermitian 对角算子的酉轨道 ${\mathcal{你}}_{\mathcal{钾}+\mathbb{C}}$ 紧凑算子的单元化 $\mathcal{钾}(\mathcal{H})+\mathbb{C}$，或等效地，商 ${\mathcal{你}}_{\mathcal{钾}+\mathbb{C}}/{\mathcal{你}}_{\mathcal{D}(\mathcal{钾}+\mathbb{C})}$. 我们将此与不同幺正子群的动作联系起来，以描述连接端点的度量测地线（使用自然距离）。因此，我们获得了局部 Hopf-Rinow 定理。我们还探讨了关于短曲线唯一性的情况，并证明存在一些不能使用最小反厄米算子参数化的短曲线$\mathcal{钾}(\mathcal{H})+\mathbb{C}$.