Summary: The construction of frames for a Hilbert space \({\mathcal H}\) 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 \(\{\sqrt {c_i}\}\). 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, Kovačević, 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.
For the entire collection see [Zbl 1052.42002].