Graphs of operators as points in the Grassmann manifold
Abstract
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× H) of orthogonal projections in H× H. We show that the subset Γb of graphs of bounded operators is the open unit ball of P(H× H) centered at the graph of the zero operator P0 (which projects onto H×{0}). This ball is diffeomorphic to B(H) via the map T↦ PT (= 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× 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.
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BibTeX
@misc{andruchow2026graphs,
title = {{Graphs of operators as points in the Grassmann manifold}},
author = {Esteban Andruchow and Lazaro Recht and Alejandro Varela},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30120},
eprint = {2608.30120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30120},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Graphs of operators as points in the Grassmann manifold
AU - Esteban Andruchow
AU - Lazaro Recht
AU - Alejandro Varela
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30120
UR - https://arxiv.org/abs/2608.30120
AB - 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× H) of orthogonal projections in H× H. We show that the subset Γ^b of graphs of bounded operators is the open unit ball of P(H× H) centered at the graph of the zero operator P_0 (which projects onto H×{0}). This ball is diffeomorphic to B(H) via the map T↦ 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× 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.
ER -