# 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