Stability of Operator-Valued Truncated Moment Problems
- 1 Department of Mathematics, Politechnica University of Bucharest, Bucharest, Romania
Abstract
In this note a multidimensional Hausdorff truncated operator-valued moment problem, from the point of view of “ stability concept ” of the number of atoms of the obtained atomic, operator-valued representing measure for the terms of a finite, positively define kernel of operators, is studied. The notion of “ stability of the dimension ” in truncated, scalar moment problems was introduced in [1]. In this note, the concept of “ stability ” of the algebraic dimension of the obtained Hilbert space from the space of the polynomials of finite, total degree with respect to the null subspace of a unital square positive functional, in [1], is adapted to the concept of stability of the algebraic dimension of the Hilbert space obtained as the separated space of some space of vectorial functions with respect to the null subspace of a hermitian square positive functional attached to a positive definite kernel of operators. In connection with the stability of the dimension of such obtained Hilbert space, a Hausdorff truncated operator-valued moment problem and the stability of the number of atoms of the representing measure for the terms of the given operator kernel, in this note, is studied.
- F. H. Vasilescu, “Dimension Stability in Truncated Moment Problems,” Journal of Mathematical Analysis and Applications, Vol. 388, No. 1, 2012, pp. 219-230. doi:10.1016/j.jmaa.2011.11.063
- R. E. Curto and L. A. Fialkow, “Truncated K-Moment Problems in Several Variables,” Journal Operator Theory, Vol. 54, No. 1, 2005, pp. 189-226.
- R. E. Curto and L. A. Fialkow, “Flat Extension of Positive Moment Matrices. Relation in Analytic or Conjugate Terms,” Operator Theory Advanced and Applications, Vol. 104, Birkhäuser Verlag, Basel, 1998, pp. 59-82.
- T. Ando, “Truncated Moment Problems for Operators,” Acta Scientia Mathematica, Szeged, Vol. 31, 1970, pp. 319-334.
- M. Putinar, “Inverse Problems of Perturbation Theory and Moment Problems,” Functional Analysis and Related Topics, World Scientific, Singapore City, 1991, pp. 99-116.
- M. Putinar and F. H. Vasilescu, “Solving Moment Problems by Dimensional Extension,” Annals of Mathematics, Vol. 148, No. 3, 1999, pp. 1087-1107. doi:10.2307/121083
- L. Lemnete-Ninulescu, “Truncated Trigonometric and Hausdorff Moment Problems for Operators,” An Operator Theory Summer, Proceedings of the 23th International Operator Conference, Timisoara, 29 June-4 July 2010, Theta, 2012, pp. 51-61.