Completion of Hankel partial contractions of extremal type (Q2872430)

For given real numbers \(a_1,a_2,\dots,a_n\), let \(H_{\triangle}(a_1,a_2,\dots,a_n)\) denote an upper triangular Hankel matrix with first row equal to \([ a_1,a_2,\dots,a_n]\) and with undetermined entries in the lower triangular part. The matrix \(H_{\triangle}(a_1,a_2,\dots,a_n)\) is said to be a partial contraction if each of its well-defined submatrices is a contraction, and it is called extremal if \(a_1^2+a_2^2+\dots+a_n^2=1\).NEWLINENEWLINEIn this article, the authors consider a problem of finding a contractive completion of the upper triangular extremal Hankel matrix when \(n=4\), that is of the matrix \(H_{\triangle}(a,b,c,d)\). They give a necessary condition and some sufficient conditions on \(a,b,c\) and \(d\) to guarantee that the contractive completion of the upper triangular extremal Hankel matrix exists.NEWLINENEWLINEAs an application of their results, the authors give necessary and sufficient conditions for the upper triangular Hankel matrix \(H_{\triangle}(a,b,c,x)\), with given \(a,b\) and \(c\), to be extremal and to have a contractive completion.