×

Completion of non-negative block operators in Banach spaces. (English) Zbl 0939.47002

Let \({\mathcal B}\) be the direct sum of Banach spaces \({\mathcal B}_i\) \((i= 1,2,3)\), which is also a Banach space. A bounded linear operator \(A:{\mathcal B}\to{\mathcal B}^*\), where \({\mathcal B}^*\) is the Banach space of bounded antilinear functionals on \({\mathcal B}\), has a block representation \(A= (A_{jk})^3_{j,k}\), where \(A_{jk}:{\mathcal B}_k\to{\mathcal B}^*_j\) \((j,k= 1,2,3)\).
The completion problem solved by the authors is: for given \(A_{j,k}\) \((j,k= 1,2)\) and \(A_{lm}\) \((l,m= 2,3)\) determine the set of all operators \(A_{13}\) and \(A_{31}\) such that \(A\) is a nonnegative operator (i.e. the functional \(Ax\geq 0\) for all \(x\in{\mathcal B}\)).
This is a generalization of a finite-dimensional problem. A particular truncated trigonometric moment problem is also solved by means of the above result.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A57 Linear operator methods in interpolation, moment and extension problems
PDFBibTeX XMLCite
Full Text: DOI