Li, Chunji

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Title ON THE MOMENT MATRIX $E(n)$ (Current topics onoperator theory and operator inequalities)Author(s)Li, ChunjiCitation数理解析研究所講究録 (2002), 1259: 60-65Issue Date2002-04URL http://hdl.handle.net/2433/41966RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityON THE MOMENT MATRIX $E$ $(n)$Chunji LiYanbian University, P. R. China1 Introduction and preliminariesConsider acollection of complex numbers$\gamma\equiv\gamma^{(2n)}$ : $\gamma_{0},\gamma_{01},\gamma_{10}$ , $\cdots,\gamma_{0,2n},\gamma_{1,2n-1}$ , $\cdots,\gamma_{2n-1,1},\gamma_{2n,\mathrm{O}}$ ,with $\gamma\alpha \mathrm{l}$ $>0$ and $\gamma_{j:}=\overline{\gamma}_{ij}$ . The truncated complex moment problem entailsfinding apositive Borel measure $\mu$ supported in the complex plane $\mathbb{C}$ suchthat$\gamma_{\dot{l}j}=\int\overline{z}^{\dot{\iota}}z^{j}d\mu(z)$ $(0\leq i+j\leq 2n)$ ; (1.1)$\mu$ is called arepresenting measure for $\gamma$. This truncated complex momentproblem has been well-established ([CuFl], $[\mathrm{C}\mathrm{u}\mathrm{F}2]$ , [JLLL]).We recall first some notation from [CuFl] and $[\mathrm{C}\mathrm{u}\mathrm{F}2]$ . For $n\geq 1$ , let$m\equiv m(n)=(n+1)(n+2)/2$ . For $A\in \mathcal{M}_{m}(\mathbb{C})$ (the $m\cross m$ complexmatrices), we denote the successive rows and columns according to the fol-lowing lexicographic-functional ordering:$\mathit{1}$ Z $\overline{Z}$ $Z^{2}$ $\overline{Z}$Z $Z-2$ $\ldots$ $Z^{n}$ $Z-Z^{n-1}$ $\ldots$ ) $Z-_{n-1}$ Z $Z-n$ .We define $M(n):=M(n)(\gamma)\in \mathcal{M}_{m}(\mathbb{C})$ as folows: for $0\leq k+l\leq n,0\leq$$i+j\leq n$ , the entry in row $\overline{Z}^{k}Z^{l}$ and column $\overline{Z}\dot{.}Z^{j}$ is $M(n)(k,l)(:,j)=\gamma_{l+\dot{\cdot},j+k}$ .These matrices come from Bram-Halmos characterization for acyclic oper-ator $T$ satisfying $\gamma_{\dot{l}j}=(T^{*:}T^{j}x_{0},x_{0})$ , where $x_{0}$ is acyclic vector for $T$ (cf.[Br] or [Con] $)$ . So it is nature to consider moment matrices correspondedby Embry characterization for subnormality of such operators (cf. [Em], or[Con] $)$ . We $\mathrm{w}\mathrm{i}\mathrm{U}$ write such matrices by $E(n)$ .1259 2002 60-6560Consider acollection of complex numbers$\gamma\equiv\{\gamma_{ij}\}(0\leq i+j\leq 2n, |i-j|\leq n)$ with $\gamma_{\alpha 1}>0$ and $\gamma_{ji}=\overline{\gamma}_{ij}$ .For $n\in \mathrm{N}$ , let$m=m[n]=([ \frac{n}{2}]+1)$ $\{$ $[ \frac{n+1}{2}]+1)$ .For $A\in M_{m}(\mathbb{C})$ , we first introduce the following order on the rows andcolumns of $A$ : 1, $Z$, $Z^{2},\overline{Z}Z$ , $Z^{3},\overline{Z}Z^{2}$ , $Z^{4},\overline{Z}Z^{3},\overline{Z}^{2}Z^{2}$ , $Z^{5}$ , $\cdots$ . We denote theentry of $A$ in row $\overline{Z}^{k}Z^{l}$ and column $\overline{Z}^{i}Z^{j}$ by $A_{(k,l)(i,j)}$ . $\mathrm{I}\mathrm{f}n=2k$ , $k=1$ , 2, $\cdots$ ,let$SP_{n}=\{p(z,\overline{z})=a\alpha) +a_{01}z+a_{02}z^{2}+a_{11}\overline{z}z+a_{03}z^{3}+a_{12}\overline{z}z^{2}+\cdots+a_{kk}\overline{z}^{k}z^{k}\}$ ;if $n=2k+1$ , $k=0,1,2$ , $\cdots$ , let$SP_{n}=\{p(z,\overline{z})=a_{0}+a_{01}z+a_{02}z^{2}+a_{11}\overline{z}z+a_{03}z^{3}+a_{12}\overline{z}z^{2}+\cdots+a_{k,k+1}\overline{z}^{k}z^{k+1}\}$ ,where $a_{ij}\in \mathrm{C}$ . It is clear that $SP_{n}$ is asubspace of $\mathcal{P}_{n}$ , the vector space ofall complex polynomials in $z,\overline{z}$ of total degree $\leq n$ . For $p\in SP_{n}$ , let $\hat{p}=$$[\mathrm{a}\mathrm{o}\mathrm{o}, a_{01}, \cdots, a_{kk}]^{T}$ (which means the transposed) or $[a_{0}, \mathrm{a}\mathrm{o}\mathrm{i}, \cdots, a_{k,k+1}]^{T}$ in$\mathbb{C}^{m}$ . We define asesquilinear form $\langle\cdot, \cdot\rangle_{A}$ on $SP_{n}$ by $\langle p, q\rangle_{A}:=(\mathrm{A}\mathrm{p},$ $q\gamma(p,$ $q\in$$SP_{n})$ . In particular, $\langle\overline{z}^{i}z^{j},\overline{z}^{k}z^{l}\rangle_{A}=A_{(k,l)(i,j)}$ , for $0\leq i+j\leq n$ , $i\leq j$ and$0\leq k+l\leq n$ , $k\leq l$ . For 7, we define the moment matrix $E(n)\equiv E(n)(\gamma)\in$$M_{m}(\mathbb{C})$ as follows: $E(n)_{(k,l)(i,j)}:=\gamma_{l+i,j+k}$ .The following provides amotivation to study the truncated moment the-ory of $E(n)$ , whose proof can be found in [JKLP].Theorem 1.1. Let $S$ be a contractive subnormal operator $with$ a cyclicvector $x_{0}$ in $\mathcal{H}$ and let $\gamma_{\dot{\mathrm{t}}j}=(S^{*:}S^{j}x_{0}, x_{0})$ . Then the following assertionsare equivalent:(i) $M(n)\geq 0$ for any $n\in \mathrm{N}$ ;(ii) $E(n)\geq 0$ , for any $n\in \mathrm{N}$ ;(iii) there eists a positive Borel measure $\mu$ supported in the complex plane$\mathbb{C}$ such that$\gamma_{ij}=\int_{\mathrm{J}\mathrm{D}}\overline{z}^{i}z^{j}d\mu(z)$ for any $i,j\in \mathrm{N}\cup\{0\}$ ,where $\mathrm{D}$ is the closed unit disc in C.61We may give the following conjecture, as in [CuFl].Conjecture 1.2. Let $\gamma\equiv\{\gamma_{\dot{l}\mathrm{j}}\}(0\leq i+j\leq 2n, |i-j|\leq n)$ be a truncatedmoment sequence. The following statements are equivalent:(i) 7has a rank $E(n)$ -atomic representing measure;(ii) $E(n)\geq 0$ and $E(n)$ admits a flat dension $E(n+1)$ .In this article, we will consider the conjecture concretely and give thedouble flat extension theorem.2Moment matrices $E(n)$ and representing mea-suresIf $\mu$ is the representing measure for $\gamma$ , then $\langle$$E(n)\hat{p},p7$ $= \int|p(z,\overline{z})|^{2}d\mu$ , for$p(z,\overline{z})\in SP_{n}$ . Hence $E(n)\geq 0$ . But the converse implication is not alwaystrue (see Example 2.2 below). We first introduce an analogous statementwith that of $M(n)$ . For $p\in SP_{n}$ , let $Z(p)=\{z\in \mathbb{C}:\mathrm{p}\{\mathrm{z},\overline{z})=0\}$ .Lemma 2.1. ([JKLP]) Let $\gamma\equiv\{\gamma_{\dot{l}j}\}(0\leq i+j\leq 2n, |i-j|\leq n)$ .Assume that $\gamma$ has a representing measure $\mu$ . For $p\in SP_{n}$ , $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mu\subseteq$$Z(p)\Leftrightarrow p(Z,\overline{Z})=0$ .Example 2.2. Consider$M:=E(3)=[01001-i102001-i1$ $000021+i0021-i1-i1402(1+i)1+i1+i1+i$ $402(1-i)002]$ .It is easy to show that $E(3)$ is positive and rank $E(3)=3$.In fact, $\det([M]_{4})=$$\det([M]_{5})=\det M=0$ , where $[M]_{k}$ is the left upper $k\cross k$ submatrix. Fur-thermore, we have$\overline{Z}Z$$=$ $\mathit{1}+\frac{1-i}{2}Z^{2}$ ,$\overline{Z}Z^{2}$$=$ $2Z$,$Z^{3}$ $=$ $(1 +i)l+(1 +i)Z$.62$p_{1}(z,\overline{z})$ $=$ $1+ \frac{1-i}{2}z^{2}-\overline{z}z$ ,$p_{2}(z,\overline{z})$ $=$ $2z$ $-\overline{z}z^{2}$ ,$p_{3}(z,\overline{z})$ $=$ $(1+i)+(1 +i)z-z^{3}$ .Then $Z(p_{1},p_{2},h)=\{z\in \mathbb{C} : p:(z,\overline{z})=0, i=1,2,3\}=\emptyset$. Thus for thegiven moment sequence $\gamma$ in $E(3)$ , there is no representing measure for $\gamma$ .Theorem 2.3. ([CuFl]) If $\gamma$ $\equiv\{\gamma_{ij}\}(0\leq i \%$j $\leq 2n)$ is flat and$M(n)\geq 0$ , then $M(n)$ admits a unique flat extension of the fom $M(n+1)$ .The above theorem produces Conjecture 1.2.We showed this conjecture is true in the case of even numbers [JKLP].We can provide acounter example for Conjecture 1.2 in the case $n=3$.Example 2.4. (Example 2.2 revisited) Since rank $E(2)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$ $E(3)=3$,$E(3)$ is flat. If $E(3)$ admits aflat extension $E(4)$ , then$Z^{4}=$ $(1 +i)Z+(1 +i)Z^{2}$ , $\overline{Z}Z^{3}=2Z^{2}$ . (2.1)Prom the first equality of (2.1), we obtain $\gamma_{34}=2$ , and from the secondequality of (2.1), we obtain $\gamma_{34}=0$ . Hence $E(3)$ has no flat extension of$E(4)$ .So we have the following theorems in sharpness whose proof can be foundin [JKLP].Theorem 2.5. Let $n\geq 2$ . If $\gamma$ is double flat ( $i.e.$ , rank $E(n)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$$E(n-2))$ and $E(n)\geq 0$ , then $E(n)$ admits a unique flat extension of theform $E(n+1)$ .Theorem 2.6. The truncated complex moment sequence $\gamma\equiv\{\gamma_{\dot{|}j}\}(0\leq$$i+j\leq 2n$ , $|i-j|\leq n)$ has $a$ rank $E(n)$-atomic representing measure if andonly if $E(n)\geq 0$ and $E(n)$ admits a double flat extension $E(n+2)$ , $\mathrm{i}.\mathrm{e}.$ ,rank $E(n)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$ $E(n+2)$ .Finally, we give the following example that affirm Theorem 2.6.Example 2.7. Let$E(2)=\mathrm{I}10-i1$ $01+i11-i$ $3i3i1+i$ $3-3i1-i1]$63Then $E(2)$ admits adouble flat extension $E(4)$ as the following$E(4)=[_{B}E(3)B^{*}C]$ ,where$E(3)$ $=$ $[10-i11-i-1-i3011-i1+i-3i4+4ii4-4i33i1+i4-4i \frac{11}{3}3i-4-4i-ii-14+4i3i11-4+4i11i4+4i4-4i31+i11-11i]$ ,$B$ $=$ $[3-3i-34+4i4-4i-4-4i$ $11i11-11\mathrm{i}$ $11-11\mathrm{i}-11$ $15(1+i)15(1-i)-15(1-i)$ $15(1+i)15(1-i)-15(1+i)]$ ,$C=$ $\{\begin{array}{lll}4\mathrm{l} -4\mathrm{l}i -4\mathrm{l}41i 41 -41i-41 41i 41\end{array}\}$ .In fact, rank $E(2)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$ $E(4)=2$. Since$\{\begin{array}{l}z^{2}=i+(1+i)z\overline{z}z=\mathrm{l}+(1-i)z\end{array}$we obtain two atoms $\mathrm{a}$ $=$ $(1 -\sqrt{3})(1+i)/2$ and $z_{1}=(1 +\sqrt{3})(1+i)/2$ .According to$\{\begin{array}{ll}1 1z_{0} z_{1}\end{array}\}\{\begin{array}{l}\rho_{\mathrm{O}}\rho_{1}\end{array}\}=\{\begin{array}{l}\gamma\alpha]\gamma_{\mathrm{O}1}\end{array}\}$ ,we have $\rho_{0}=\frac{1+\sqrt{3}}{2\sqrt{3}}$ , $\rho_{1}=\frac{-1+\sqrt{3}}{2\sqrt{3}}$ . Thus we obtain the representing measure$\mu=\rho_{0}\delta_{z\mathrm{o}}+\rho_{1}\delta_{z_{1}}$ .References[Br] J. Bram, Subnormal operators, Duke Math. J. 22(1$55), 75-94.[Con] J. Conway, The Theory of Subnormal Operators, Mathematical Sur-veys and Monographs, Amer. Math. Soc. 36(1\Re 1).64[CuFl] R. Curto and L. Fialkow, Solution of the truncated complex momentproblems for flat data, Memoirs Amer. Math. Soc. 568(1996).[CuF2] –, Flat extensions of positive moment matrices: recursivelygenerated relations, Memoirs Amer. Math. Soc. 648(1998).[CuP] R. Curto and M. Putinar, Nearly subnormal opeators and momentproblems, J. Funct. Anal. 115(1993), 480-497.[Em] M. Embry, A generalization of the Halmos-Bram criter ion for sub-normality, Acta. Sci. Math. (Sz), 31(1973), 61-64.[JKLP] II Bong Jung, Eungil Ko, Chunji Li and Sang Soo Park, Embrytruncated complex moment problem, preprint.[JLLL] II Bong Jung, Sang Hoon Li, $\mathrm{W}\infty$ Young Lee and Chunji Li, Thequariic moment problem, submitted to TVans. of Amer. Math. Soc.65

ON THE MOMENT MATRIX $E(n)$ (Current topics on operator theory and operator inequalities)

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ON THE MOMENT MATRIX $E(n)$ (Current topics on operator theory and operator inequalities)

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ON THE MOMENT MATRIX E(n)E(n)E(n) (Current topics on operator theory and operator inequalities)

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ON THE MOMENT MATRIX $E(n)$ (Current topics on operator theory and operator inequalities)