# Matrix Completions, Moments, and Sums of Hermitian Squares

Mihály Bakonyi
Hugo J. Woerdeman
Pages: 536
https://www.jstor.org/stable/j.ctt7rp1d

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
4. Chapter One Cones of Hermitian matrices and trigonometric polynomials
(pp. 1-68)

In this chapter we study cones in the real Hilbert spaces of Hermitian matrices and real valued trigonometric polynomials. Based on an approach using such cones and their duals, we establish various extension results for positive semidefinite matrices and nonnegative trigonometric polynomials. In addition, we show the connection with semidefinite programming and include some numerical experiments.

Let 𝓗 be a Hilbert space over ℝ with inner product 〈·,·〉. A nonempty subset 𝒞 of 𝓗 is called a cone if

(i) 𝒞 + 𝒞 ⊆ 𝒞,

(ii)α𝒞 ⊆ 𝒞 for allα> 0.

Note that a cone isconvex(i.e.,...

5. Chapter Two Completions of positive semidefinite operator matrices
(pp. 69-174)

In this chapter we are concerned with positive definite and semidefinite completions of partial operator matrices, and we consider the banded case in Section 2.1, the chordal case in Section 2.2, the Toeplitz case in Section 2.3, and the generalized banded case and the operator-valued positive semidefinite chordal case in Section 2.6. In Section 2.4 we introduce the Schur complement and use it to derive an operator-valued Fejér-Riesz factorization. Section 2.5 is devoted to describing the structure of positive semidefinite operator matrices. In Section 2.7 we study the Hamburger problem based on positive semidefinite completions of Hankel matrices. Finally, in...

6. Chapter Three Multivariable moments and sums of Hermitian squares
(pp. 175-256)

In one variable one can formulate several very closely related problems that become vastly different in the multivariable case, as follows. Given are complex numbers$c_0 = \overline{c_0} , \ldots ,c_n = \overline{c_{ - n}}$. Consider the following problems.

1.Carathéodory interpolation problem: find an analytic function ϕ:𝔻 → ℂ such that Re ϕ(z) ≥ 0,z∈ 𝔻 (that is, ϕ belongs to theCarathéodory class), ϕ(0) =c₀, and$\frac{\phi^{(k)}(0)}{k!}= 2c_{k}, k \in \{1, \ldots, n\}$.

2.Moment problem: find a positive measure σ on 𝕋 with moments$\hat \sigma (k): = {\text{ }}\int_\mathbb{T} {{z^k}d\sigma = {c_k},k \in \{ - n, \ldots ,n\} }$.

3.Bernstein-Szegő measure moment problem: find a polynomial$p(z) = \sum\nolimits_{k = 0}^n {p_k z^k }$with$p(z) \ne 0,z \in \overline{\mathbb{D}}$(p(z) isstable), so that the Fourier coefficients$\hat f(k)$of$f(z): = \frac{{p_0 }}{{|p(z)|^2 }}$satisfy$\hat f(k) = c_k ,k \in \{ - n, \ldots ,n\}$....

7. Chapter Four Contractive analogs
(pp. 257-360)

An operator$T \in \mathcal{L}(\mathcal{H},\mathcal{H}')$is called acontractionif ∥T∥ ≤ 1, and astrict contractionif ∥T∥ < 1. In this chapter we deal with contractive completions of partial operator matrices. Since the norm of a submatrix is always less or equal to the norm of the matrix itself, every partial matrix which admits a contractive completion has to bepartially contractive(or apartial contraction), that is, all its fully specified submatrices are contractions.

A contractive completion problem can always be reduced to a positive semidefinite one. That is because Lemma 2.4.4 implies that ∥T∥ ≤ 1 if and only...

8. Chapter Five Hermitian and related completion problems
(pp. 361-474)

In this chapter we consider various completion problems that are in one way or another closely related to positive semidefinite or contractive completion problems. For instance, as a variation on requiring that all eigenvalues of the completion are positive/nonnegative, one can consider the question how many eigenvalues of a Hermitian completion have to be positive/nonnegative. In the solution to the latter problem ranks of off-diagonal parts will play a role, which is why we also discuss minimal rank completions. Related is a question on real measures on the real line. As a variation of the contractive completion problem, we will...

9. Bibliography
(pp. 475-512)
10. Subject Index
(pp. 513-516)
11. Notation Index
(pp. 517-518)
12. Back Matter
(pp. 519-520)