发表机构
Instituto Argentino de Matemática, ‘Alberto P. Calderón’, CONICET; Universidad Nacional de General Sarmiento（阿根廷数学研究所，Alberto P. Calderón，CONICET; 圣马丁国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将希尔伯特空间中闭稠定算子的图视为格拉斯曼流形的子集，分析有界/无界算子图的性质、自伴算子图间的极小测地线，还关联紧算子图与受限格拉斯曼流形并研究公共补问题。
AI 中文摘要
我们研究希尔伯特空间H中闭稠定算子的图构成的集合Γ，将其视为H×H上正交投影构成的格拉斯曼流形P(H×H)的子集。我们证明，有界算子的图构成的子集Γᵇ是P(H×H)中以零算子图P₀（投影到H×{0}）为中心的开单位球，该球通过映射T↦P_T（即投影到T的图Gr(T)）与B(H)微分同胚。我们还证明，无界闭算子的图位于Γ的边界上。此外，我们研究连接两个图Gr(A)、Gr(B)的P(H×H)的极小测地线的存在性与特征：若A、B为自伴算子，此类测地线始终存在，我们利用与子空间对Gr(A)、Gr(B)相关的五空间分解显式构造出一个特殊指数；一个低维实例表明，连接两个图的测地线未必留在Γ内，即并非完全由图构成。我们还将紧算子的图与受限格拉斯曼流形关联，研究Gr(S)、Gr(T)对的公共补问题，当其中一个算子有界或满足下有界条件时给出肯定结果。
英文摘要
We study the set $Γ$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\times H)$ of orthogonal projections in $H\times H$. We show that the subset $Γ^b$ of graphs of bounded operators is the open unit ball of $P(H\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\times\{0\}$). This ball is diffeomorphic to $B(H)$ via the map $T\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $Γ$. We also study the existence and characteristics of minimal geodesics of $P(H\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $Γ$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.
