Cayley transform
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) |
Cayley transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley_transform&oldid=46290