The Complex Moment Problem and Subnormality: A Polar Decomposition Approach

Abstract

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. Academic PressIt is well known that there is a relationship between moment problems and Hilbert space operators, one of the most beautiful examples of this kind. In particular there is a link between the complex moment problem and cyclic subnormal operators (cf. [16], for instance), where any progress in one side impacts the other side. While the bounded case is pretty well understood, in the unbounded one a search for satisfactory solutions is still required. In this paper we provide a solution of the complex moment problem and, in a parallel way, a characterization of unbounded subnormal operators which are not necessarily cyclic. Though the latter case includes the former, we have decided to separate them, thus giving the possibility of a choice to readers with particular interests.Let us be more precise: A bisequence of complex numbers indexed by integer lattice points of the first quarter of the plane may be positive or positive definite. While positivity is necessary and sufficient for the bisequence to be a complex moment one, it requires information on how non-negative polynomials in two real variables look like and this is not article no. FU983284 432

The truncated complex $K$-moment problem

“…Concrete conditions (in terms of the β i ) for the existence of such a flat extension are given in [Fi1]. In [CuFi1] we proved that β (2n) has a representing measure supported in [a, b] if and only if there is an r-atomic representing measure supported in [a, b], and that this occurs if and only if H (n) ≥ 0 admits a flat extension for which bH (n) ≥ H x (n) ≥ aH (n); Theorem 5.5 thus refines this result in the case when r ≤ n. [PuVa2], [Rez], [StSz1], [StSz2], [Va1], [Va2]). For instance, a rather concrete Riesz-type condition is obtained in [PuVa1], when K is a compact semi-algebraic set.…”

Operators with absolute continuity properties: an application to quasinormality

“…Recall that subnormal (hyponormal) operators are closable and their closures are subnormal (hyponormal). We refer the reader to [15,7,19] and [20,21,22,23] for elements of the theory of unbounded hyponormal and subnormal operators, respectively.…”

Unbounded subnormal weighted shifts on directed trees

“…It has been developed in two main directions, the first is purely theoretical (cf. [39,30,61,18,69,15,16,17,70,71,2]), the other is related to special classes of operators (cf. [13,34,35,36]).…”

“…Using the above characterization, we show that some weak-type limit procedure preserves subnormality (this can also be done with the help of either [58,Theorem 3] or [61,Theorem 37]; however these two characterizations take more complicated forms). This is a key tool for proving Theorem 5.2.1.…”