Diagonality and idempotents with applications to problems in operator theory and frame theory

TL;DR

This paper characterizes zero-diagonal idempotent operators and explores their applications in operator and frame theory, including diagonalization properties and dual frame inner products.

Contribution

It provides new characterizations of zero-diagonal idempotents and links these to problems in operator and frame theory, including diagonal representations and dual frame inner products.

Findings

A nonzero idempotent is zero-diagonal iff it is not a Hilbert-Schmidt perturbation of a projection.

Any bounded sequence can be realized as the diagonal of some idempotent operator.

Any absolutely summable sequence with positive integer sum can be the diagonal of a finite rank idempotent.

Abstract

We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis. We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.