mjo-cone.tex

   1 %
   2 % Cone stuff.
   3 %
   4 % The operator families Z(K), LL(K), etc. can technically be defined on
   5 % sets other than cones, but nobody cares.
   6 %
   7
   8 \usepackage{amssymb} % \succcurlyeq and friends
   9
  10 \input{mjo-common}
  11
  12 % The dual of a subset of an inner-product space; always a closed
  13 % convex cone.
  14 \newcommand*{\dual}[1]{ #1^{*} }
  15
  16 %
  17 % Common cones.
  18 %
  19
  20 % The nonnegative orthant in the given number of dimensions.
  21 \newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
  22
  23 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
  24 \newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
  25
  26 % The PSD cone in a space of symmetric matrices.
  27 \newcommand*{\Snplus}[1][n]{ \mathcal{S}^{{#1}}_{+} }
  28
  29 % The PSD cone in a space of Hermitian matrices.
  30 \newcommand*{\Hnplus}[1][n]{ \mathcal{H}^{{#1}}_{+} }
  31
  32
  33 %
  34 % Some collections of linear operators.
  35 %
  36
  37 % The set of all positive operators on its argument. This uses the
  38 % same magic as \boundedops to accept either one or two arguments. If
  39 % one argument is given, the domain and codomain are equal and the
  40 % positive operators fix a subset of that space. When two arguments
  41 % are given, the positive operators send the first argument to a
  42 % subset of the second.
  43 \newcommand*{\posops}[2][]{
  44   \pi\of{ {#2}
  45     \if\relax\detokenize{#1}\relax
  46       {}%
  47     \else
  48       {,{#1}}%
  49     \fi
  50   }
  51 }
  52
  53 % The set of all S-operators on its argument.
  54 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
  55
  56 % The cone of all Z-operators on its argument.
  57 \newcommand*{\Zof}[1]{ \mathbf{Z} \of{ {#1} } }
  58
  59 % The space of Lyapunov-like operators on its argument.
  60 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
  61
  62
  63 %
  64 % Cone inequality operators.
  65 %
  66
  67 % Standard cone inequalities.
  68 \newcommand*{\gek}{\succcurlyeq}
  69 \newcommand*{\gtk}{\succ}
  70 \newcommand*{\lek}{\preccurlyeq}
  71 \newcommand*{\ltk}{\prec}
  72
  73
  74 % Starred versions of the cone inequality operators.
  75 \newcommand*{\ineqkstar}[1]{ \mathrel{ \overset{ _{\ast} }{ #1 } } }
  76 \newcommand*{\gekstar}{ \ineqkstar{\gek} }
  77 \newcommand*{\gtkstar}{ \ineqkstar{\gtk} }
  78 \newcommand*{\lekstar}{ \ineqkstar{\lek} }
  79 \newcommand*{\ltkstar}{ \ineqkstar{\ltk} }
  80
  81 % And negated versions of some of those...
  82 \newcommand*{\ngeqkstar}{ \ineqkstar{\nsucceq} }
  83 \newcommand*{\ngtrkstar}{ \ineqkstar{\nsucc} }