Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

ANDRUCHOW, ESTEBAN; CHIUMIENTO, EDUARDO; VARELA, ALEJANDRO

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2021 vol. 500

0022-247X

Let H be a reproducing kernel Hilbert space of functions on a set X. We studythe problem of finding a minimal geodesic of the Grassmann manifold of H thatjoins two subspaces consisting of functions which vanish on given finite subsets ofX. We establish a necessary and sufficient condition for existence and uniquenessof geodesics, and we then analyze it in examples. We discuss the relation of thegeodesic distance with other known metrics when the mentioned finite subsets aresingletons. We find estimates on the upper and lower eigenvalues of the uniqueself-adjoint operators which define the minimal geodesics, which can be made moreprecise when the underlying space is the Hardy space. Also for the Hardy space wediscuss the existence of geodesics joining subspaces of functions vanishing on infinitesubsets of the disk, and we investigate when the product of projections onto thistype of subspaces is compact.