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Absolute Extreme Points of Matrix Convex Sets


Let $\bbS (\cH_n)^g$ denote $g$-tuples of self-adjoint operators on a Hilbert space $\cH_n$ with $\dim \cH_n=n$. Given tuples $X=(X_1, \dots, X_g) \in \bbS (\cH_{n_1})^g$ and $Y=(Y_1, \dots, Y_g) \in \bbS (\cH_{n_2})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form


V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n


where $V_1:\cH_n \to \cH_{n_1}$ and $V_2: \cH_{n} \to \cH_{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the $g$-tuples $X, Y$, and $V_1^* XV_1+V_2^* Y V_2$ do not need to have the same size. As a result, matrix convex sets are a dimension-free analog of convex sets.

While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary.

A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations.

Chapter \ref{ch:MatConvWOAbs} shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets we consider are noncommutative sets, $K_X$, formed by taking matrix convex combinations of a single tuple $X$. If $X$ is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces and $0$ is in the finite interior of $K_X$, then $K_X$ is a closed bounded matrix convex set with no absolute extreme points.

In Chapter \ref{ch:ArvSpanSpec}, we show that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are a minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals.

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