18 research outputs found
An Application of the Schur Complement to Truncated Matricial Power Moment Problems
The main goal of this paper is to reconsider a phenomenon which was treated
in earlier work of the authors' on several truncated matricial moment problems.
Using a special kind of Schur complement we obtain a more transparent insight
into the nature of this phenomenon. In particular, a concrete general principle
to describe it is obtained. This unifies an important aspect connected with
truncated matricial moment problems
On the structure of Hausdorff moment sequences of complex matrices
The paper treats several aspects of the truncated matricial
-Hausdorff type moment problems. It is shown that each
-Hausdorff moment sequence has a particular intrinsic
structure. More precisely, each element of this sequence varies within a closed
bounded matricial interval. The case that the corresponding moment coincides
with one of the endpoints of the interval plays a particular important role.
This leads to distinguished molecular solutions of the truncated matricial
-Hausdorff moment problem, which satisfy some extremality
properties. The proofs are mainly of algebraic character. The use of the
parallel sum of matrices is an essential tool in the proofs.Comment: 53 pages, LaTeX; corrected typos, added missing notation
On the Matricial Truncated Moment Problem. II
We continue the study of truncated matrix-valued moment problems begun in
arXiv:2310.00957. Let . Suppose that
is a measurable space and is a
finite-dimensional vector space of measurable mappings of into
, the Hermitian matrices. A linear functional
on is called a moment functional if there exists a
positive -valued measure on
such that for
.
In this paper a number of special topics on the truncated matricial moment
problem are treated. We restate a result from (Mourrain and Schm\"udgen, 2016)
to obtain a matricial version of the flat extension theorem. Assuming that
is a compact space and all elements of are
continuous on we characterize moment functionals in terms of
positivity and obtain an ordered maximal mass representing measure for each
moment functional. The set of masses of representing measures at a fixed point
and some related sets are studied. The class of commutative matrix moment
functionals is investigated. We generalize the apolar scalar product for
homogeneous polynomials to the matrix case and apply this to the matricial
truncated moment problem