Generating operator of a semi-group

The derivative at zero of a semi-group of linear operators (cf. Semi-group of operators) $ T ( t) $, $ 0 < t < \infty $, acting on a complex Banach space $ X $. If $ T ( t) $ is continuous in the operator norm, then it has the form $ T ( t) = e ^ {t A _ {0} } $, where $ A _ {0} $ is a bounded operator,

$$ \tag{1 } \lim\limits _ {t \rightarrow 0 } \frac{T ( t) x - x }{t} = A _ {0} x $$

for any $ x \in X $, and $ A _ {0} $ is the generating operator of this semi-group. Conversely, if the limit on the left-hand side exists, for all $ x \in X $, then $ T ( t) = e ^ {t A _ {0} } $.

A more complicated situation arises when $ T ( t) $ is only a strongly-continuous semi-group. In this case the limit (1) does not necessarily exist for every $ x $. The operator $ A _ {0} $, defined on the linear set $ D ( A _ {0} ) $ of all $ x $ for which the limit exists, is linear and unbounded, and is called the infinitesimal operator. In particular, $ A _ {0} $ is defined on all elements of the form $ \int _ \alpha ^ \beta T ( t) y dt $, $ \alpha , \beta > 0 $, $ y \in X $. If $ X _ {0} $ denotes the closure of the union of the range of values of all $ T ( t) $, $ t > 0 $, then $ D ( A _ {0} ) $ is dense in $ X _ {0} $ and, moreover, $ \cap _ {n} D ( A _ {0} ^ {n} ) $ is dense in $ X _ {0} $. The values of $ A _ {0} $ also lie in $ X _ {0} $. If $ A _ {0} $ is an unbounded operator, then $ D ( A _ {0)} $ is a set of the first category in $ X _ {0} $.

If $ X _ {0} $ does not contain elements $ x $ for which $ T ( t) x \equiv 0 $, then $ A _ {0} $ has a closure $ A = \overline{ {A _ {0} }}\; $, which is also called the generating operator of the semi-group $ T ( t) $. In this case, for $ x \in D ( A) $,

$$ \tag{2 } T ( t) x - T ( s) x = \int\limits _ { s } ^ { t } T ( \tau ) Ax d \tau , $$

$$ \frac{dT ( t) x }{dt} = A _ {0} T ( t) x = T ( t) Ax . $$

These equations define an operator $ A $ which is, generally speaking, an extension of the closure of $ A _ {0} $. It is also called the generalized generating operator of $ T ( t) $.

On the set $ D _ {R} $ of all $ x \in X $ for which the improper integral

$$ \tag{3 } \int\limits _ { 0 } ^ { t } T ( s) x ds $$

converges, one defines the operator

$$ R ( \lambda ) x = \lim\limits _ {t \rightarrow 0 } \int\limits _ { t } ^ \infty e ^ {- \lambda s } T ( s) x ds $$

for $ \mathop{\rm Re} \lambda > \omega $, where $ \omega $ is the type of the semi-group $ T ( t) $. This operator has the following properties:

1) $ R ( \lambda ) D _ {R} \subset D _ {R} $;

2) $ R ( \lambda ) x - R ( \mu ) x = ( \mu - \lambda ) R ( \lambda ) R ( \mu ) x $;

3) $ R ( \lambda ) ( \lambda I - A _ {0} ) x = x $, $ x \in D ( A _ {0} ) $;

4) $ ( \lambda I - A ) R ( \lambda ) x = x $, $ x \in D _ {R} \cap X _ {0} $.

If the integral (3) converges absolutely for any $ x \in X $, then the generating operator $ A $ exists if and only if $ T ( t) x \equiv 0 $, $ x \in X $, implies $ x = 0 $; the operator $ R ( \lambda ) $ is bounded and, if $ X = X _ {0} $, it coincides with the resolvent of $ A $. For $ A _ {0} $ to be closed (i.e. for $ A = A _ {0} $) it is necessary and sufficient that

$$ \lim\limits _ {t \rightarrow 0 } \frac{1}{t} \int\limits _ { 0 } ^ { t } T ( s) x ds = x $$

for any $ x \in X _ {0} $.

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 $ R ( \lambda ) $.

References

[1]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501
[2]	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 MR264459
[3]	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 MR278121

References

[a1]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701
[a2]	A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) MR0710486 Zbl 0516.47023