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

$$ A = \ \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm cotan} \ { \frac{s}{2} } dF _ {s} . $$

References

[1]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[2]	B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French)