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Cayley transform

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of a linear (dissipative) operator with domain dense in a Hilbert space

The operator , defined on the subspace . A matrix version of this transform was considered by A. Cayley. The Cayley transform establishes a correspondence between the properties of operators with spectra "close" to the real line and operators with almost-unitary spectra (close to the circle ). The following propositions are true: 1) if is a linear dissipative operator, then is a contraction (i.e. , ) and ; 2) if is a contraction, and is dense in , then for some linear dissipative operator ; in fact, ; 3) is symmetric (self-adjoint) if and only if is isometric (unitary); 4) , where , in particular, is bounded if and only if ; and 5) if is an operator ideal in , then implies ; if are bounded operators, then the converse is also valid: implies . The Cayley transform also establishes a correspondence between certain other characteristics of the operators and : classifications of parts of the spectrum, multiplicities of spectra, structures of invariant subspaces, functional calculi, spectral decompositions, etc. Thus, if is a self-adjoint operator with resolution of the identity , , then , for , is a resolution of the identity for and

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)
How to Cite This Entry:
Cayley transform. N.K. Nikol'skii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cayley_transform&oldid=12556
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098