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gitweb.michael.orlitzky.com - mjotex.git/blob - mjo-cone.tex
4 % The operator families Z(K), LL(K), etc. can technically be defined on
5 % sets other than cones, but nobody cares.
8 \usepackage{amssymb
} % \succcurlyeq and friends
12 % The dual of a subset of an inner-product space; always a closed
14 \newcommand*
{\dual}[1]{ {#1}^
{*
} }
20 % The nonnegative orthant in the given number of dimensions.
21 \newcommand*
{\Rnplus}[1][n
]{ \Rn[#1]_
{+
} }
23 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
24 \newcommand*
{\Lnplus}[1][n
]{ \mathcal{L
}^
{{#1}}_
{+
} }
26 % The PSD cone in a space of symmetric matrices.
27 \newcommand*
{\Snplus}[1][n
]{ \mathcal{S
}^
{{#1}}_
{+
} }
29 % The PSD cone in a space of Hermitian matrices.
30 \newcommand*
{\Hnplus}[1][n
]{ \mathcal{H
}^
{{#1}}_
{+
} }
34 % Some collections of linear operators.
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][]{
45 \if\relax\detokenize{#1}\relax
53 % The set of all S-operators on its argument.
54 \newcommand*
{\Sof}[1]{ \mathbf{S
} \of{ {#1} } }
56 % The cone of all Z-operators on its argument.
57 \newcommand*
{\Zof}[1]{ \mathbf{Z
} \of{ {#1} } }
59 % The space of Lyapunov-like operators on its argument.
60 \newcommand*
{\LL}[1]{ \mathbf{LL
}\of{ {#1} } }
64 % Cone inequality operators.
67 % Standard cone inequalities.
68 \newcommand*
{\gek}{\succcurlyeq}
69 \newcommand*
{\gtk}{\succ}
70 \newcommand*
{\lek}{\preccurlyeq}
71 \newcommand*
{\ltk}{\prec}
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} }
81 % And negated versions of some of those...
82 \newcommand*
{\ngeqkstar}{ \ineqkstar{\nsucceq} }
83 \newcommand*
{\ngtrkstar}{ \ineqkstar{\nsucc} }