Admissible sequences of positive operators

Abstract:

A scalar sequence $\xi$ is said to be admissible for a positive operator $A$ if $A= \sum \xi _jP_j$ for some rank-one projections $P_j$, or, equivalently, if diag $\xi$ is the diagonal of $VAV^*$ for some partial isometry $V$ having as domain the closure of the range of $A$. The main result of this paper is that if $A$ is the sum of infinitely many projections (converging in the strong operator topology) and $\xi$ is a nonsummable sequence in $[0,1]$ that satisfies the Kadison condition that requires that either $\sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}+ \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} = \infty$ or the difference $\sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}- \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\}$ is an integer, then $\xi$ is admissible for $A$. This result extends Kadison’s carpenter’s theorem and provides an independent proof of it.