Completion of Non-negative Block Operators in Banach Spaces

Completion of Non-negative Block Operators in Banach Spaces Let the Banach space $$\mathcal{B}$$ be the direct sum of three Banach spaces $$\mathcal{B}_1 ,{\text{ }}\mathcal{B}_2 ,$$ and $$\mathcal{B}_3$$ . A bounded linear operator $$A$$ of $$\mathcal{B}$$ to the Banach space $$\mathcal{B}^*$$ of bounded antilinear functionals on $$\mathcal{B}$$ can be represented in the block form $$A = (A_{jk} )_{j,k = 1}^3$$ , where $$A_{jk}$$ is a bounded linear operator of $$\mathcal{B}_k$$ to $$\mathcal{B}_j^*$$ . Using a generalization of a Lemma due to Efimov and Potapov (see [4, Section 4 of $$\S$$ 2]) we solve the following completion problem: Assume that the operators $$A_{jk} ,j,k = 1,2,$$ and $$A_{lm} ,l,m = 2,3$$ , are given. Describe the sets of all operators $$A_{13}$$ and $$A_{31}$$ , such that the operator $$A$$ is non-negative, i.e., that for all $$x \in \mathcal{B}$$ the value of the functional $$Ax$$ taken at $$x$$ is non-negative. Our description is a generalization of the result in the finite-dimensional case (see [3, Theorem 3.2.1]). As its consequence we will obtain a solution to a certain truncated trigonometric moment problem. Positivity Springer Journals

Completion of Non-negative Block Operators in Banach Spaces

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Kluwer Academic Publishers
Copyright © 1999 by Kluwer Academic Publishers
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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