Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
نویسندگان
چکیده
منابع مشابه
Matrix Representations for Positive Noncommutative Polynomials
In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ”symmetric” polyno...
متن کاملOn a convex operator for finite sets
Let S be a finite set with n elements in a real linear space. Let JS be a set of n intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved that each homothet of conv S that is contained in aff S can be obtained using this operator. A vari...
متن کاملNoncommutative Polynomials Nonnegative on a Variety Intersect a Convex Set
By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D◦ L of a linear matrix inequality (LMI) L(X) 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called “Perfect” Positivstellensatz. For example, given a generic convex fre...
متن کاملConvex Noncommutative Polynomials Have Degree Two or Less
A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided p(tX + (1 − t)Y ) ≤ tp(X) + (1 − t)p(Y ) for all 0 ≤ t ≤ 1 and for all tuples X = (X1, . . . , Xg) and Y = (Y1, . . . , Yg) of symmetric matrices on a common finite dimensional vector space of a sufficiently large dimension (depending upon p). The main result of this paper is that every matrix convex p...
متن کاملFaber Polynomials of Matrices for Non-convex Sets
It has been recently shown that ||Fn(A)|| ≤ 2, where A is a linear continuous operator acting in a Hilbert space, and Fn is the Faber polynomial of degree n corresponding to some convex compact E ⊂ C containing the numerical range of A. Such an inequality is useful in numerical linear algebra, it allows for instance to derive error bounds for Krylov subspace methods. In the present paper we ext...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2017
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2016.07.043