On the completely indeterminate case for block Jacobi matrices

We consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.

Multivariable moments and sums of Hermitian squares

This chapter shows how five closely related problems become vastly different when one considers the multivariable case. In a similar way, various natural sums of Hermitian squares problems that one may consider in the multivariable case all come down to the classical factorization problem in one variable as discussed in Section 1.1. The discussions cover positive Carathéodory interpolation on the polydisk; inverses of multivariable Toeplitz matrices and Christoel–Darboux formulas; two-variable moment problem for Bernstein–Szeg ő measures; Fejéer–Riesz factorization and sums of Hermitian squares; completion problems for positive semidefinite functions on amenable groups; moment problems on free groups, noncommutative factorization; two-variable Hamburger moment problem; and Bochner's theorem and an application to autoregressive stochastic processes. Exercises and notes are provided at the end of the chapter.