Asymptotically Optimal Multi-Paving
| Autor: | Ravichandran, Mohan, Srivastava, Nikhil |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions $A^{(1)}, \cdots, A^{(k)} \in M_n(\mathbb{C})$ and $\epsilon > 0$, one may find a paving $X_1 \amalg \cdots \amalg X_r = [n]$ where $r \leq 18k\epsilon^{-2}$ such that, \[\lambda_{max} (P_{X_i} A^{(j)} P_{X_i}) < \epsilon, \quad i \in [r], \, j \in [k].\] As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be $(O(\epsilon^{-2}),\epsilon)$ paved. As an application, we give a simplified proof wth slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices. Comment: 23 pages. In the previous version, we had erroneously claimed that the main theorem in this paper implies a polylogarithmic bound in the commutator theorem of Johnson, Ozawa and Schechtman. This has been corrected with a weaker bound. The main results in the paper are unchanged |
| Databáze: | arXiv |
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