+\newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
+
+% An inline column vector, with parentheses and a transpose operator.
+\newcommand*{\colvec}[1]{ \transpose{\left({#1}\right)} }
+
+% Bounded linear operators on some space. The required argument is the
+% domain of those operators, and the optional argument is the
+% codomain. If the optional argument is omitted, the required argument
+% is used for both.
+\newcommand*{\boundedops}[2][]{
+ \mathcal{B}\of{ {#2}
+ \if\relax\detokenize{#1}\relax
+ {}%
+ \else
+ {,{#1}}%
+ \fi
+ }
+}
+
+
+%
+% Orthogonal direct sum.
+%
+% First declare my ``perp in a circle'' operator, which is meant to be
+% like an \obot or an \operp except has the correct weight circle. It's
+% achieved by overlaying an \ocircle with a \perp, but only after we
+% clip off the top half of the \perp sign and shift it up.
+\DeclareMathOperator{\oplusperp}{\mathbin{
+ \ooalign{
+ $\ocircle$\cr
+ \raisebox{0.625\height}{$\clipbox{0pt 0pt 0pt 0.5\height}{$\perp$}$}\cr
+ }
+}}
+
+% Now declare an orthogonal direct sum in terms of \oplusperp.
+\newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
+