Completion of non-negative block operators in Banach spaces

Let the Banach space B be the direct sum of three Banach spaces B-1, B-2, a nd B-3. A bounded linear operator A of B to the Banach space B* of bounded antilinear functionals on B can be represented in the block form A = (A(jk) )(j,k=1)(3), where A(jk) is a bounded linear operator of B-k to B-j*. Using a generalization of a Lemma due to Efimov and Potapov (see [4, Section 4 o f 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 a ll operators A(13) and A(31), such that the operator A is non-negative, i.e ., that for all x is an element of B the value of the functional Ax taken a t x is non-negative. Our description is a generalization of the result in t he finite-dimensional case (see [3, Theorem 3.2.1]). As its consequence we will obtain a solution to a certain truncated trigonometric moment problem.