Wavelet-Packet Content for Positive Operators

We study positive operator decompositions associated with rooted trees of orthogonal projections. In this sense, the refinement tree induces an ``MRA in $B\left(H\right)_{+}$''. To each node we assign a positive content operator, and these contents split along the tree and yield a positive decomposition at each fixed depth. The resulting decomposition gives a multiresolution description of positive operators adapted to the tree. In the trace class setting, the scalar contents determine a canonical boundary measure on the path space, and for each vector the corresponding quadratic data admit a nonnegative integrable density with respect to that measure. At fixed depth, we study greedy extraction rules based on trace and Hilbert-Schmidt norm. The trace rule gives a sharp geometric decay estimate for the trace of the positive remainder. In the Hilbert-Schmidt setting, a depth dependent coherence parameter measures departure from block diagonal form and yields geometric decay bounds. We also study adaptive partitions up to a terminal depth. In that setting, the change in total squared content under local refinement is determined by off-diagonal interaction among the child contents. This leads to an additive refinement calculus for adaptive decompositions and recursive criteria for optimal adaptive partitions.