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…
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…
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…
Duality, extreme points and hulls for noncommutative partial convexity
Igor Klep, Scott McCullough, Tea Štrekelj
This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of -convexity. Here is a tuple of free symmetric…
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 $(…