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:
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 , ;
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.
|||E.B. Davies, "One-parameter semigroups" , Acad. Press (1980)|
|||B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French)|
|[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=51663