Some new equivalences of Anderson’s paving conjectures

Paulsen, Vern; Raghupathi, Mrinal

doi:10.1090/S0002-9939-08-09644-5

Some new equivalences of Anderson’s paving conjectures
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by Vern I. Paulsen and Mrinal Raghupathi

Proc. Amer. Math. Soc. 136 (2008), 4275-4282

DOI: https://doi.org/10.1090/S0002-9939-08-09644-5

Published electronically: July 22, 2008

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Abstract:

Anderson’s paving conjectures are known to be equivalent to the Kadison-Singer problem. We prove some new equivalences of Anderson’s conjectures that require the paving of smaller sets of matrices. We prove that if the strictly upper triangular operators are paveable, then every $0$ diagonal operator is paveable. This result follows from a new paving condition for positive operators. In addition, we prove that if the upper triangular Toeplitz operators are paveable, then all Toeplitz operators are paveable.

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