Admissible sequences for positive operators

TL;DR

This paper extends Kadison's carpenter theorem by establishing sufficient conditions for admissible sequences for positive operators, including sums of projections, providing new insights into operator diagonals.

Contribution

It generalizes Kadison's theorem to sums of projections, offering an alternative proof and expanding understanding of admissible sequences for positive operators.

Findings

Sufficient condition for admissibility when A is a sum of projections

Independent proof of Kadison's carpenter theorem

Extension of admissibility criteria to broader classes of positive operators

Abstract

A sequence of scalars is said to be admissible for a positive operator A on a Hilbert space if it is the diagonal of VAV* for some partial isometry V having as domain the closure of the range of A. When A is a projection, the celebrated Kadison's carpenter theorem provides a sufficient condition for a sequence to be admissible for A. We prove that the same condition is sufficient for the sequence to be admissible for A when A is a sum of projections (converging in the SOT). This provides an independent proof of Kadison's carpenter theorem.