X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo-cone.tex;h=6c681e9ebee4a7d0a9b7fef673ac3bf45cd1f1d5;hb=d85706cc1efea350df9e957f6583a92d4e05359f;hp=443d7009af0a757b2897341259da2134f360fb90;hpb=58238ffe26be32382a6b9cc6aada94b9badc5170;p=mjotex.git

diff --git a/mjo-cone.tex b/mjo-cone.tex
index 443d700..6c681e9 100644
--- a/mjo-cone.tex
+++ b/mjo-cone.tex
@@ -9,6 +9,10 @@
 
 \input{mjo-common}
 
+% The dual of a subset of an inner-product space; always a closed
+% convex cone.
+\newcommand*{\dual}[1]{ #1^{*} }
+
 %
 % Common cones.
 %
@@ -27,9 +31,25 @@
 
 
 %
-% 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} } }
 
@@ -39,16 +59,18 @@
 % 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 }
+\newcommand*{\gek}{\succcurlyeq}
+\newcommand*{\gtk}{\succ}
+\newcommand*{\lek}{\preccurlyeq}
+\newcommand*{\ltk}{\prec}
 
 
 % Starred versions of the cone inequality operators.