  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.
