  This paper studies shift operators which arises from extractions of solutions
for Lasserre's hierarchy. First, we establish a connection between
multiplication operators and shift operators. More importantly, we derive new
positive semidefinite conditions of rank-one moment sequences via shift
operators, and utilize these conditions to strengthen Lasserre's hierarchy for
real and complex polynomial optimization. Furthermore, we integrate the
strengthening technique with correlative sparsity and sign symmetries present
in polynomial optimization problems. Extensive numerical experiments show that
our strengthening technique can significantly improve the bound (especially for
complex polynomial optimization) and allows to achieve global optimality at
lower relaxation orders, thus providing substantial computational savings and
considerable speedup.
