  The construction of frames for a Hilbert space H can be equated to the
decomposition of the frame operator as a sum of positive operators having rank
one. This realization provides a different approach to questions regarding
frames with particular properties and motivates our results. We find a
necessary and sufficient condition under which any positive finite-rank
operator B can be expressed as a sum of rank-one operators with norms specified
by a sequence of positive numbers {c_i}. Equivalently, this result proves the
existence of a frame with B as it's frame operator and with vector norms given
by the square roots of the sequence elements. We further prove that, given a
non-compact positive operator B on an infinite dimensional separable real or
complex Hilbert space, and given an infinite sequence {c_i} of positive real
numbers which has infinite sum and which has supremum strictly less than the
essential norm of B, there is a sequence of rank-one positive operators, with
norms given by {c_i}, which sum to B in the strong operator topology.
  These results generalize results by Casazza, Kovacevic, Leon, and Tremain, in
which the operator is a scalar multiple of the identity operator (or
equivalently the frame is a tight frame), and also results by Dykema, Freeman,
Kornelson, Larson, Ordower, and Weber in which {c_i} is a constant sequence.
