Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:500

Article

Andruchow, Esteban^1,3  Chiumiento, Eduardo^2,3  Varela, Alejandro^1,3 

[1] Univ Nacl Gral, Inst Ciencias, Sarmiento, Los Polvorines, Argentina

[2] UNLP, Ctr Matemdt, Fac Ciencias Exactas, La Plata, Argentina

[3] Consejo Nacl Invest Cient & Tecn, Inst Argentine Matemat Alberto P Calderon, Buenos Aires, DF, Argentina

关键词: Geodesics;    Grassmann manifold;    Reproducing kernels;    Analytic functions spaces;    Zero sets;    Hardy space;   

DOI  :  10.1016/j.jmaa.2021.125107

来源: Elsevier

【 摘 要 】

Let Hbe a reproducing kernel Hilbert space of functions on a set X. We study the problem of finding a minimal geodesic of the Grassmann manifold of Hthat joins two subspaces consisting of functions which vanish on given finite subsets of X. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free