\input{mjo-common}
+% The dual of a subset of an inner-product space; always a closed
+% convex cone.
+\newcommand*{\dual}[1]{ #1^{*} }
+
%
% Common cones.
%
% The nonnegative orthant in the given number of dimensions.
-\newcommand*{\Rnplus}[1][n]{ \mathbb{R}^{{#1}}_{+} }
+\newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
% The Lorentz ``ice-cream'' cone in the given number of dimensions.
\newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
%
-% Some collections of linear operators
+% Some collections of linear operators.
%
+% The set of all positive operators on its argument. This uses the
+% same magic as \boundedops to accept either one or two arguments. If
+% one argument is given, the domain and codomain are equal and the
+% positive operators fix a subset of that space. When two arguments
+% are given, the positive operators send the first argument to a
+% subset of the second.
+\newcommand*{\posops}[2][]{
+ \pi\of{ {#2}
+ \if\relax\detokenize{#1}\relax
+ {}%
+ \else
+ {,{#1}}%
+ \fi
+ }
+}
+
% The set of all S-operators on its argument.
\newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
% The space of Lyapunov-like operators on its argument.
\newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
+% The Lyapunov rank of the given cone.
+\newcommand*{\lyapunovrank}[1]{ \beta\of{ {#1} } }
-%
% Cone inequality operators.
-%
-
-% Standard cone inequalities.
-\newcommand*{\gek}{ \succcurlyeq }
-\newcommand*{\gtk}{ \succ }
-\newcommand*{\lek}{ \preccurlyeq }
-\newcommand*{\ltk}{ \prec }
-
-
-% Starred versions of the cone inequality operators.
-\newcommand*{\ineqkstar}[1]{ \mathrel{ \overset{ _{\ast} }{ #1 } } }
-\newcommand*{\gekstar}{ \ineqkstar{\gek} }
-\newcommand*{\gtkstar}{ \ineqkstar{\gtk} }
-\newcommand*{\lekstar}{ \ineqkstar{\lek} }
-\newcommand*{\ltkstar}{ \ineqkstar{\ltk} }
-
-% And negated versions of some of those...
-\newcommand*{\ngeqkstar}{ \ineqkstar{\nsucceq} }
-\newcommand*{\ngtrkstar}{ \ineqkstar{\nsucc} }
+\newcommand*{\gecone}{\succcurlyeq}
+\newcommand*{\gtcone}{\succ}
+\newcommand*{\lecone}{\preccurlyeq}
+\newcommand*{\ltcone}{\prec}