# Cayley transform

of a linear (dissipative) operator $A$ with domain $\mathop{\rm Dom} A$ dense in a Hilbert space $H$
The operator $C _ {A} = ( A - iI) ( A + iI) ^ {-} 1$, defined on the subspace $\mathop{\rm Dom} C _ {A} = ( A + iI) \mathop{\rm Dom} A$. A matrix version of this transform was considered by A. Cayley. The Cayley transform establishes a correspondence between the properties of operators $A$ with spectra $\sigma ( A)$" close" to the real line and operators with almost-unitary spectra (close to the circle $\{ {\zeta \in \mathbf C } : {| \zeta | = 1 } \}$). The following propositions are true: 1) if $A$ is a linear dissipative operator, then $C _ {A}$ is a contraction (i.e. $\| C _ {A} x \| \leq \| x \|$, $x \in \mathop{\rm Dom} A$) and $\mathop{\rm Ker} ( I - C _ {A} ) = \{ 0 \}$; 2) if $T$ is a contraction, $\mathop{\rm Ker} ( I - T) = \{ 0 \}$ and $( I - T) \mathop{\rm Dom} T$ is dense in $H$, then $T = C _ {A}$ for some linear dissipative operator $A$; in fact, $A = i ( I + T) ( I - T) ^ {-} 1$; 3) $A$ is symmetric (self-adjoint) if and only if $C _ {A}$ is isometric (unitary); 4) $\sigma ( A) = \omega ( \sigma ( C _ {A} ))$, where $\omega ( \zeta ) = i ( 1 + \zeta ) ( 1 - \zeta ) ^ {-} 1$, in particular, $A$ is bounded if and only if $1 \notin \sigma ( C _ {A} )$; and 5) if $\gamma$ is an operator ideal in $H$, then $A - B \in \gamma$ implies $C _ {A} - C _ {B} \in \gamma$; if $A, B$ are bounded operators, then the converse is also valid: $C _ {A} - C _ {B} \in \gamma$ implies $A - B \in \gamma$. The Cayley transform also establishes a correspondence between certain other characteristics of the operators $A$ and $C _ {A}$: classifications of parts of the spectrum, multiplicities of spectra, structures of invariant subspaces, functional calculi, spectral decompositions, etc. Thus, if $A$ is a self-adjoint operator with resolution of the identity $\{ E _ {t} \}$, $t \in \mathbf R$, then $\{ F _ {s} \}$, $F _ {s} = E _ {t}$ for $s = - 2 \mathop{\rm arctan} t$, is a resolution of the identity for $C _ {A}$ and
$$A = \ \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm cotan} \ { \frac{s}{2} } dF _ {s} .$$