Contraction semi-group

A one-parameter strongly-continuous semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258903.png" />, of linear operators in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258904.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258905.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258906.png" /> that is densely defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258907.png" /> is a generating operator (generator) of the contraction semi-group if and only if the Hille–Yosida condition is satisfied:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258908.png" />

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c0258909.png" />. In other words, a densely-defined operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589011.png" />, unitary semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589012.png" />, self-adjoint semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589013.png" /> and normal semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589014.png" />. Instead of the generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589015.png" /> it is sometimes convenient to use its Cayley transform: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589016.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589017.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589018.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589019.png" />. In order that a contraction semi-group is completely non-unitary it is sufficient that it be stable, that is, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589020.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589021.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589022.png" />.

For every contraction semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589023.png" /> there is an orthogonal decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589024.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589025.png" />-invariant subspaces such that the semi-group is unitary on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589026.png" /> and completely non-unitary on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589027.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589028.png" /> is a contraction semi-group in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589029.png" />, then there is a larger Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589030.png" />, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589031.png" /> as a subspace, and in it a unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589033.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589034.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589036.png" /> is the orthogonal projection from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589037.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589038.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589039.png" /> is called a unitary dilation of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589040.png" />. The dilation is uniquely defined up to an isomorphism if it is required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589041.png" /> coincides with the closed linear span of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589042.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589043.png" />) (a minimal dilation).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589044.png" /> be a Hilbert space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589045.png" /> be the Hilbert space of all measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589046.png" />-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589048.png" />, with square-integrable norm. In this space, the unitary group of two-sided shifts, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589049.png" />, is defined. Similarly, the semi-group of one-sided shifts is defined in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589051.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589052.png" />

Every completely non-unitary semi-group of isometries is isomorphic to the one-sided shift on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589053.png" /> for some suitable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589054.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589055.png" /> is a completely non-unitary contraction semi-group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589056.png" /> is its minimal unitary dilation, then on some invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589057.png" /> (but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589058.png" /> is stable, then on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025890/c02589059.png" />) 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)

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)