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Markov-type and operator-valued multidimensional moment problems, with some applications. (English) Zbl 1174.47011

Let \(T\) be a measurable space, \(\nu\) be a positive \(\sigma\)-finite measure on \(T\), and \(X:=L_{\nu}^{1}(T)\) be endowed with the usual norm \(\left\| x\right\|_{1}=\int_{T}\left| x(t)\right| dv(t),\;x\in X,\) and let \(X_{+}:=\{x\in X:\;x(t)\geq0 \;\nu\text{-a.e. on } T\}\). For an arbitrary family \(J\) of subscripts, let \(\left\{ x_{j},j\in J\right\} \subset X\) and \(\left\{ y_{j},j\in J\right\} \subset\mathbb{R}.\)
A Markov-type problem in this setting is the following: on the set \(\left\{ y_{j},j\in J\right\}\), find necessary and sufficient conditions for the existence of a function \(h\in L_{\nu}^{\infty}(T)\) with the properties \(-1\leq h(t)\leq1\) \(\nu\text{-a.e. on } T\) and \(\int_{T}x_{j}(t)h(t)\,d\nu(t)=y_{j}\), \(j\in J.\) The authors find the following sufficient condition: for any subset \(J_{0}\subset J\) and any\(\left\{ \lambda_{j},j\in J_{0}\right\} \subset\mathbb{R}\), we have \[ \sum_{i,j\in J_{0}}\lambda_{i}\lambda_{j}y_{1}y_{j}\leq\sum_{i,j\in J_{0} }\lambda_{i}\lambda_{j}\cdot\int_{T}x_{i}(t)\,d\nu(t)\cdot\int_{T}x_{j}(t)\,d\nu(t). \] The result is applied to obtain a sufficient condition on \(y_{j}\), \(j\in \mathbb{Z}_{+}^{2}\), for the existence of a solution of the Markov classical two-dimensional moment problem on an unbounded non-semialgebraic subset of \(\mathbb{R}^{2},\) and to a Markov moment problem on an arc of a hyperbola. A construction of the solution \(h\) for a Markov moment problem on the ellipse is also obtained. Finally, some classical Markov-type problems on closed polydiscs in \(\mathbb{C}^{n}\) are considered. In this case, the solution \(F\) maps some space \(X\) of analytic functions on \(\mathbb{C}^{n}\) into a space \(Y\) of selfadjoint commuting operators, acting on a Hilbert space \(H.\) Some examples in \(L^{2}\left( \left[ 0,\frac{\pi}{2}\right] \right) \) and \(L^{2}\left( \left[ 0,\infty\right] \right) \) are given.

MSC:

47A57 Linear operator methods in interpolation, moment and extension problems
44A60 Moment problems
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