Some new equivalences of Anderson's paving conjectures

TL;DR

This paper establishes new equivalences for Anderson's paving conjectures, linking the pavability of specific classes of operators to the general conjecture, thus advancing understanding of the Kadison-Singer problem.

Contribution

It introduces new conditions under which certain classes of operators imply the pavability of all operators, simplifying the approach to Anderson's conjectures.

Findings

Pavability of strictly upper triangular operators implies pavability of all 0 diagonal operators.

A new paving condition for positive operators is established.

Pavability of upper triangular Toeplitz operators implies pavability of all Toeplitz operators.

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 operatorss are pavable, then every 0 diagonal operator is pavable. 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.