Flat extensions of nonsingular moment matrices (Q1039553)

For a given closed subset \(K\subset\mathbb{C}\) and a finite double sequence \(\gamma=\{\gamma_{ij}\}_{0\leq i+j\leq 2n}\) of complex numbers with \(\gamma_{00}>0\) and \(\gamma_{ji}=\bar{\gamma_{ij}}\;(0\leq i+j\leq 2n)\), the truncated \(K\)-moment problem entails finding a positive Borel measure \(\mu\) (called a representing measure for \(\gamma\)) such that its support is contained in \(K\) and \[ \gamma_{ij}=\int\bar{z}^iz^j\,d\mu(z),\quad 0\leq i+j\leq 2n. \] Using the ordering \[ 1,\;Z,\;\bar{Z},\;Z^2,\;\bar{Z}Z,\;\bar{Z}^2,\;Z^3,\;\bar{Z}Z^2,\;\bar{Z}^2Z,\;\bar{Z}^3, \dots \] for its successive rows and columns, the corresponding moment matrix \(M(n)\equiv M(n)(\gamma)\) is defined by the entries \(M(n)_{(k,l)(i,j)}=\gamma_{i+l,j+k}\) on the row \(\bar{Z}^kZ^l\) and the column \(\bar{Z}^iZ^j\). \textit{L.\,A.\thinspace Fialkow} proposed in [Contemp.\ Math.\ 185, 133--150 (1995; Zbl 0830.44007)] the following conjecture: if \(M(n)\) is positive and invertible, then \(M(n)\) has a flat extension \(M(n+1)\). This conjecture is true for \(n=1\) [\textit{R.\,E.\thinspace Curto} and \textit{L.\,A.\thinspace Fialkow}, Mem.\ Am.\ Math.\ Soc.\ 568 (1996; Zbl 0876.30033)] and false for \(n=3\) [Operator Theory: Adv.\ Appl.\ 104, 59--82 (1998; Zbl 0904.30020)]. It is the aim of the paper under review to propose some partial solutions regarding the existence of flat extensions for nonsingular quartic moment matrices (the case \(n=2\)). More precisely, the authors consider the cases \(M(1)=I\) and \(\gamma_{12}=\gamma_{03}=0\); \(M(1)=I\), \(\gamma_{12}=\gamma_{13}=\gamma_{04}=0\) and \(\gamma_{03}\neq 0\); respectively, \(M(1)=I\), \(\gamma_{12}=\gamma_{13}=0\) and \(\gamma_{03}\gamma_{04}\neq 0\). In each of these situations, certain sufficient conditions are given in order to ensure that \(M(2)\) has a flat extension \(M(3)\).