  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.
