View source for Generating operator of a semi-group

The derivative at zero of a semi-group of linear operators (cf. [[Semi-group of operators|Semi-group of operators]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439202.png" />, acting on a complex Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439203.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439204.png" /> is continuous in the operator norm, then it has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439206.png" /> is a bounded operator,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439208.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g0439209.png" /> is the generating operator of this semi-group. Conversely, if the limit on the left-hand side exists, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392011.png" />.

A more complicated situation arises when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392012.png" /> is only a [[Strongly-continuous semi-group|strongly-continuous semi-group]]. In this case the limit (1) does not necessarily exist for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392013.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392014.png" />, defined on the linear set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392015.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392016.png" /> for which the limit exists, is linear and unbounded, and is called the infinitesimal operator. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392017.png" /> is defined on all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392021.png" /> denotes the closure of the union of the range of values of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392024.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392025.png" /> and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392026.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392027.png" />. The values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392028.png" /> also lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392030.png" /> is an unbounded operator, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392031.png" /> is a set of the first category in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392032.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392033.png" /> does not contain elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392034.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392036.png" /> has a closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392037.png" />, which is also called the generating operator of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392038.png" />. In this case, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392039.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392041.png" /></td> </tr></table>

These equations define an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392042.png" /> which is, generally speaking, an extension of the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392043.png" />. It is also called the generalized generating operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392044.png" />.

On the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392045.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392046.png" /> for which the improper integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

converges, one defines the operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392048.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392050.png" /> is the type of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392051.png" />. This operator has the following properties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392052.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392053.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392055.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392057.png" />.

If the integral (3) converges absolutely for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392058.png" />, then the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392059.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392061.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392062.png" />; the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392063.png" /> is bounded and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392064.png" />, it coincides with the [[Resolvent|resolvent]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392065.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392066.png" /> to be closed (i.e. for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392067.png" />) it is necessary and sufficient that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392068.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392069.png" />.

The basic problem in the theory of operator semi-groups is to establish relations between properties of a semi-group and properties of its generating operator, where the latter are usually formulated in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043920/g04392070.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.P. Zabreiko, A.V. Zafievskii, "On a certain class of semigroups" ''Soviet Math. Dokl.'' , '''10''' : 6 (1969) pp. 1523–1526 ''Dokl. Akad. Nauk'' , '''189''' : 5 (1969) pp. 934–937 {{MR|264459}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.V. Zafievskii, "On semigroups with singularities summable with a power-weight at zero" ''Soviet Math. Dokl.'' , '''11''' : 6 (1970) pp. 1408–1411 ''Dokl. Akad. Nauk'' , '''195''' : 1 (1970) pp. 24–27 {{MR|278121}} {{ZBL|}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.G. Krein, "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) {{MR|0710486}} {{ZBL|0516.47023}} </TD></TR></table>