Rank-One Decomposition of Operators and Construction of Frames

Abstract

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.