Contraction semi-group
A one-parameter strongly-continuous semi-group ,
,
, of linear operators in a Banach space
for which
. An operator
that is densely defined in
is a generating operator (generator) of the contraction semi-group if and only if the Hille–Yosida condition is satisfied:
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for all . In other words, a densely-defined operator
is a generator of a contraction semi-group if and only if is a maximal dissipative operator.
Contraction semi-groups in Hilbert space have been studied in detail. Particular forms of contraction semi-groups are semi-groups of isometries , unitary semi-groups
, self-adjoint semi-groups
and normal semi-groups
. Instead of the generator
it is sometimes convenient to use its Cayley transform:
(a cogenerator). It turns out that a semi-group is a semi-group of isometries, or a unitary, a self-adjoint, or a normal semi-group if and only if the cogenerator is, respectively, an isometric, a unitary, a self-adjoint, or a normal operator.
A contraction semi-group is called completely non-unitary, if its restriction to any invariant subspace is not unitary. For a completely non-unitary semi-group as
, for any
. In order that a contraction semi-group is completely non-unitary it is sufficient that it be stable, that is, that
as
, for
.
For every contraction semi-group there is an orthogonal decomposition
into
-invariant subspaces such that the semi-group is unitary on
and completely non-unitary on
.
If is a contraction semi-group in a Hilbert space
, then there is a larger Hilbert space
, containing
as a subspace, and in it a unitary group
,
, such that
for
, where
is the orthogonal projection from
onto
. The group
is called a unitary dilation of the semi-group
. The dilation is uniquely defined up to an isomorphism if it is required that
coincides with the closed linear span of the set
(
) (a minimal dilation).
Let be a Hilbert space and let
be the Hilbert space of all measurable
-valued functions
,
, with square-integrable norm. In this space, the unitary group of two-sided shifts,
, is defined. Similarly, the semi-group of one-sided shifts is defined in the space
,
;
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Every completely non-unitary semi-group of isometries is isomorphic to the one-sided shift on for some suitable space
.
If is a completely non-unitary contraction semi-group and
is its minimal unitary dilation, then on some invariant subspace of
(but if
is stable, then on the whole of
) the group is isomorphic to that of two-sided shifts. For contraction semi-groups with non-linear operators, see Semi-group of non-linear operators.
References
[1] | E.B. Davies, "One-parameter semigroups" , Acad. Press (1980) |
[2] | B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French) |
Comments
References
[a1] | E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) |
[a2] | A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) |
Contraction semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contraction_semi-group&oldid=15504