Extending positive definiteness

The main result of the paper gives criteria for extendibility of sesquilinear form-valued mappings defined on symmetric subsets of *-semigroups to positive definite ones. By specifying this we obtain new solutions of:

• the truncated complex moment problem,
• the truncated multidimensional trigonometric moment problem,
• the truncated two-sided complex moment problem, as well as characterizations of unbounded subnormality and criteria for the existence of unitary power dilation.