Editing Cayley transform (section)

== Operator map ==
An infinite-dimensional version of an [[inner product space]] is a [[Hilbert space]], and one can no longer speak of [[matrix (mathematics)|matrices]]. However, matrices are merely representations of [[linear operator]]s, and these can be used. So, generalizing both the matrix mapping and the complex plane mapping, one may define a Cayley transform of operators.{{sfn | Rudin | 1991 | p=356-357 §13.17}}
:<math>\begin{align}
 U &{}= (A - \mathbf{i}I) (A + \mathbf{i}I)^{-1} \\
 A &{}= \mathbf{i}(I + U) (I - U)^{-1}
\end{align}</math>
Here the domain of ''U'', dom&nbsp;''U'', is (''A''+'''i'''''I'')&nbsp;dom&nbsp;''A''. See [[self-adjoint operator#Extensions of symmetric operators|self-adjoint operator]] for further details.