  We show that, for Hankel matrices, total nonnegativity (resp. total
positivity) of order r is preserved by sum, Hadamard product, and Hadamard
power with real exponent t \ge r-2. We give examples to show that our results
are sharp relative to matrix size and structure (general, symmetric or Hankel).
Some of these examples also resolve the Hadamard critical-exponent problem for
totally positive and totally nonnegative matrices.
