Search
Search Results
-
-
Theorems of Schmüdgen and Putinar
In 1991, as a corollary to a theorem about the multi-dimensional moment problem, Schmüdgen [1] proved that if a basic closed semialgebraic set...
-
The Fuglede theorem and some intertwining relations
In this paper, we show new versions of the Fuglede theorem in an unbounded setting. A related counterexample is also presented. In the second part of...
-
The Dimension One Case
In this chapter we look at certificates of positivity for polynomials that are positive on basic closed semialgebraic sets in...
-
Representation of positive polynomials on a generalized strip and its application to polynomial optimization
We study the representation of nonnegative polynomials in two variables on a certain class of unbounded closed basic semi-algebraic sets (which are...
-
Recent Developments
In this short chapter, a number of important developments and advances are summarized that mostly occurred after the 1989 publication of Einführung...
-
On the effective Putinar’s Positivstellensatz and moment approximation
We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar’s Positivstellensatz...
-
Factoring non-negative operator valued trigonometric polynomials in two variables
It is shown using Schur complement techniques that on dimensional Hilbert spaces, a non-negative operator valued trigonometric polynomial in two...
-
Positivity on Polytopes
In this chapter we discuss certificates of positivity for polytopes, which are compact basic closed semialgebraic sets in...
-
Commutativity
We have already introduced the concept of commutativity of a bounded operator with an unbounded one. In the case of two densely defined operators A...
-
The Noncompact Case
The theorems of Schmüdgen and Putinar discussed in Chap. 7 give very satisfying answers to the question of...
-
Positivity on Semialgebraic Sets
Writing a psd polynomial \(f \in \mathbb R[\underline{X}]\)...
-
Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
A noncommutative (nc ) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra...
-
Positivstellensätze for polynomial matrices
In this paper we establish some Positivstellensätze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for...
-
The moment problem on curves with bumps
The power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel,...
-
Sum-of-squares chordal decomposition of polynomial matrix inequalities
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the...
-
Spectrahedral Shadows
In this chapter we introduce the notion of a spectrahedral shadow, and examine these sets thoroughly. They are still feasible sets for semidefinite...
-
Certifying the Global Optimality of Quartic Minimization over the Sphere
The quartic minimization over the sphere is an NP-hard problem in the general case. There exist various methods for computing an approximate solution...
-
Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials
Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One...
-
