Matrix convexity

Let $V$ be a complex vector space. A set $K$ of matrices over $V$ consists of subsets $K_n\subset M_n(V)$ for all $n\in{\mathbb{N}}$. Subsets $K_1\subset V$ can be considered as sets of matrices over $V$ by putting $K_n=\emptyset$ for $n\geqslant 2$. More generally, if for some $n\in N$ the corresponding set $K_n$ is not given a priori, put this $K_n$ as the empty set. For sets of matrices, we have the following notions of convexity:⁶⁰

A set $K$ of matrices over $V$ is called matrix convex or a matrix convex set [Wit84b] if for all $x\in K_n$ and $y\in K_m$

$$x\oplus y\in K_{n+m}$$   ,

and for all $x\in K_n$ and $\alpha\in M_{n,m}$ with $\alpha^*\alpha=\mathrm{1\!\!\!\:l}_m$

$$\alpha^*x\alpha\in K_m$$   .

A set $K$ of matrices over $V$ is called absolutely matrix convex [EW97a] if for all $x\in K_n$ and $y\in K_m$

$$x\oplus y\in K_{n+m}$$   ,

and for all $x\in K_n$, $\alpha\in M_{n,m}$ and $\beta\in M_{m,n}$ with $\Vert\alpha\Vert$, $\Vert\beta\Vert\leqslant 1$

$$\alpha x\beta\in K_m$$   .

A set $K$ of matrices over $V$ is called a matrix cone [Pow74] if for all $x\in K_n$ and $y\in K_m$

$$x\oplus y\in K_{n+m}$$   ,

and for all $x\in K_n$ and $\alpha\in M_{n,m}$

$$\alpha^*x\alpha\in K_m$$   .

A set $K$ of matrices over $V$ is matrix convex if and only if all matrix convex combinations of elements of $K$ are again in $K$. $K$ is absolutely matrix convex if and only if all absolutely matrix convex combinations of elements of $K$ are again in $K$. Here, a matrix convex combination of $x_1$, ..., $x_n$ ( $x_i\in K_{k_i}$) is a sum of the form $\sum_{i=1}^n\alpha_i^*x_i\alpha_i$ with matrices $\alpha_i\in M_{k_i,j}$ such that $\sum_{i=1}^n\alpha_i^*\alpha_i=\mathrm{1\!\!\!\:l}_j$. An absolutely matrix convex combination of $x_1$, ..., $x_n$ is a sum of the form $\sum_{i=1}^n\alpha_i x_i\beta_i$ with matrices $\alpha_i\in M_{j,k_i}$ and $\beta_i\in M_{k_i,j}$ such that $\sum_{i=1}^n\alpha_i\alpha_i^*\leqslant\mathrm{1\!\!\!\:l}_j$ and $\sum_{i=1}^n\beta_i^*\beta_i\leqslant\mathrm{1\!\!\!\:l}_j$.

If $V$ is a topological vector space, topological terminology is to be considered at all matrix levels: For instance, a set $K$ of matrices over $V$ is called closed if all $K_n$ are closed.

Footnotes

... convexity:⁶⁰

With the notation $M(V)=\bigcup\{M_n(V)\;\vert\;n\in{\mathbb{N}}\}$, a set $K$ of matrices over $V$ is simply a subset of $M(V)$. The sets $K_n$ can be regained as $K_n=K\cap M_n(V)$.

Then the definitions have the following form:

Let $K\subset M(V)$. $K$ is called matrix convex if

$$K\oplus K\subset K$$   $$\mbox{ and $\alpha^*K\alpha\subset K$\ for all matrices $\alpha$\ with $\alpha^*\alpha=\mathrm{1\!\!\!\:l}$.}$$

$K$ is called absolutely matrix convex if

$$K\oplus K\subset K$$   $$\mbox{ and $\alpha K\beta\subset K$\ for all matrices $\alpha$, $\beta$\ with $\Vert\alpha\Vert$, $\Vert\beta\Vert\leqslant 1$.}$$

$K$ is called a matrix cone if

$$K\oplus K\subset K$$   $$\mbox{ and $\alpha^*K\alpha\subset K$\ for all matrices $\alpha$.}$$