Projections and idempotents with fixed diagonal and the homotopy problem for unit tight frames

TL;DR

This paper explores the structure of certain operators with fixed diagonal entries and applies these findings to resolve parts of conjectures regarding the connectedness of unit-norm tight frames.

Contribution

It provides new insights into the topology of idempotent operators with fixed diagonals and addresses open conjectures in frame theory.

Findings

Characterized the topological structure of idempotents with fixed diagonals

Resolved specific cases of conjectures on the connectedness of unit-norm tight frames

Established new links between operator theory and frame theory

Abstract

We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a fixed orthonormal basis of a Hilbert space. As an application, we settle some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of the set of unit-norm tight frames.