Wavelet-Packet Content for Positive Operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917476169678848 |
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| author | Song, Myung-Sin Tian, James |
| author_facet | Song, Myung-Sin Tian, James |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14174 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wavelet-Packet Content for Positive Operators Song, Myung-Sin Tian, James Functional Analysis Primary 47B65, Secondary 41A65, 42C40, 47A63, 47B10, 60G42 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. |
| title | Wavelet-Packet Content for Positive Operators |
| topic | Functional Analysis Primary 47B65, Secondary 41A65, 42C40, 47A63, 47B10, 60G42 |
| url | https://arxiv.org/abs/2601.14174 |