## Multivariable Operator Theory

Crow 206

#### Start Date

7-19-2016 2:30 PM

#### End Date

19-7-2016 2:50 PM

#### Description

Let $K$ denote a nonempty closed subset of $\mathbb{R}^{n}$, let $m=2d$, and let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of finite degree $m$. %and let $K$ denote a closed subset of $\mathbb{R}^{n}$. \textit{ The Truncated $K$-Moment Problem} concerns the existence of a positive Borel measure $\mu$, supported in $K$, such that $$\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).$$ The \textit{core variety} of $\beta$, $\mathcal{V} \equiv \mathcal{V}(\beta)$, is an algebraic variety in $\mathbb{R}^{n}$ that contains the support of any such \textit{$K$-representing measure}. In previous work we showed, conversely, that if $\mathcal{V}$ is a nonempty compact set, or $\mathcal{V}$ is nonempty and is a determining set for polynomials of degree at most $m$ (in particular, if $\mathcal{V}= \mathbb{R}^{n}$), then $\beta$ admits a $\mathcal{V}$-representing measure. We describe some additional cases where a nonempty core variety implies the existence of a representing measure.

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Jul 19th, 2:30 PM Jul 19th, 2:50 PM

The core variety of a multisequence in the truncated moment problem

Crow 206

Let $K$ denote a nonempty closed subset of $\mathbb{R}^{n}$, let $m=2d$, and let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of finite degree $m$. %and let $K$ denote a closed subset of $\mathbb{R}^{n}$. \textit{ The Truncated $K$-Moment Problem} concerns the existence of a positive Borel measure $\mu$, supported in $K$, such that $$\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).$$ The \textit{core variety} of $\beta$, $\mathcal{V} \equiv \mathcal{V}(\beta)$, is an algebraic variety in $\mathbb{R}^{n}$ that contains the support of any such \textit{$K$-representing measure}. In previous work we showed, conversely, that if $\mathcal{V}$ is a nonempty compact set, or $\mathcal{V}$ is nonempty and is a determining set for polynomials of degree at most $m$ (in particular, if $\mathcal{V}= \mathbb{R}^{n}$), then $\beta$ admits a $\mathcal{V}$-representing measure. We describe some additional cases where a nonempty core variety implies the existence of a representing measure.