6 papers
Beurling Criteria for Reproducing Kernels
Shuaibing Luo, Scott McCullough, Georgios Tsikalas
A classical theorem due to Beurling-Lax-Halmos characterizes the invariant subspaces of the unilateral shift as ranges of isometric multiplication operators acting on the Hardy spa…
Operator-Valued Positivstellensätze on Matrix Convex Sets and Free Products of Finite Abelian Groups
Abhay Jindal, Igor Klep, Scott McCullough
We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let be an operator-valued polynomial in $B(H…
Positive operator-valued noncommutative polynomials are squares
Abhay Jindal, Igor Klep, Scott McCullough
We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann.~Math., 2002) and one of the authors (Line…
Fejér--Riesz factorization for positive noncommutative trigonometric polynomials
Igor Klep, Jacob Levenson, Scott McCullough
We prove a Fejér-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on , where is either the…
The complete Pick property for pairs of kernels and Shimorin's factorization
Scott McCullough, Georgios Tsikalas
Let be a pair of Hilbert function spaces with kernels . In a 2005 paper, Shimorin showed that a certain factorization condition on $(…
Duality, extreme points and hulls for noncommutative partial convexity
Igor Klep, Scott McCullough, Tea Å trekelj
This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of -convexity. Here is a tuple of free symmetri…