The Complex Moment Problem and Subnormality: A Polar Decomposition Approach

It has been known that positive definiteness does not guarantee a bisequence to be a complex moment. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. To strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context (Theorems 9 and 29). All this enables us to prove characterizations of subnormality of unbounded operators having invariant domain (Theorems 37 and 39) and their further applications (Theorems 41 and 43) and a description of the complex moment problem on real algebraic curves (Theorems 52 and 56). The latter question is completed in the Appendix, in which we relate the complex moment problem to the two-dimensional real one, with emphasis on real algebraic sets.