Diagonality and idempotents with applications to problems in operator   theory and frame theory
  We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis.   We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent. 
Jireh Loreaux and Gary Weiss
math.FA