# Semi-group of operators

A family $\{ T \}$ of operators on a Banach space or topological vector space with the property that the composite of any two operators in the family is again a member of the family. If the operators $T$ are "indexed" by elements of some abstract semi-group $\mathfrak A$ and the binary operation of the latter is compatible with the composition of operators, $\{ T \}$ is known as a representation of the semi-group $\mathfrak A$. The most detailed attention has been given to one-parameter semi-groups (cf. One-parameter semi-group) of bounded linear operators on a Banach space $X$, which yield a representation of the additive semi-group of all positive real numbers, i.e. families $T ( t)$ with the property

$$T ( t + \tau ) x = T ( t) T ( \tau ) x ,\ t , \tau > 0 ,\ x \in X .$$

If $T ( t)$ is strongly measurable, $t > 0$, then $T ( t)$ is a strongly-continuous semi-group; this will be assumed in the sequel.

The limit

$$\omega = \lim\limits _ {t \rightarrow \infty } \ t ^ {-} 1 \mathop{\rm ln} \| T ( t) \|$$

exists; it is known as the type of the semi-group. The functions $T ( t) x$ increase at most exponentially.

An important characteristic is the infinitesimal operator (infinitesimal generator) of the semi-group:

$$A _ {0} x = \lim\limits _ {t \rightarrow 0 } t ^ {-} 1 [ T ( t) x - x ] ,$$

defined on the linear set $D ( A _ {0} )$ of all elements $x$ for which the limit exists; the closure, $A$, of this operator (if it exists) is known as the generating operator, or generator, of the semi-group. Let $X _ {0}$ be the subspace defined as the closure of the union of all values $T ( t) x$; then $D ( A _ {0} )$ is dense in $X _ {0}$. If there are no non-zero elements in $X _ {0}$ such that $T ( t) x \equiv 0$, then the generating operator $A$ exists. In the sequel it will be assumed that $X _ {0} = X$ and that $T ( t) x \equiv 0$ implies $x = 0$.

The simplest class of semi-groups, denoted by $C _ {0}$, is defined by the condition: $T ( t) x \rightarrow x$ as $t \rightarrow 0$ for any $x \in X$. This is equivalent to the condition: The function $\| T ( t) \|$ is bounded on any interval $( 0 , a ]$. In that case $T ( t)$ has a generating operator $A = A _ {0}$ whose resolvent $R ( \lambda , A ) = ( A - \lambda I ) ^ {-} 1$ satisfies the inequalities

$$\tag{1 } \| R ^ {n} ( \lambda , A ) \| \leq M ( \lambda - \omega ) ^ {-} n ,\ \ n = 1 , 2 , . . . ; \ \lambda > \omega ,$$

where $\omega$ is the type of the semi-group. Conversely, if $A$ is a closed operator with domain of definition dense in $X$ and with a resolvent satisfying (1), then it is the generating operator of some semi-group $T ( t)$ of class $C _ {0}$ such that $\| T ( t) \| \leq M e ^ {\omega t }$. Condition (1) is satisfied if

$$\| R ( \lambda , A ) \| \leq ( \lambda - \omega ) ^ {-} 1$$

(the Hill–Yosida condition). If, moreover, $\omega = 0$, then $T ( t)$ is a contraction semi-group: $\| T ( t) \| \leq 1$.

A summable semi-group is a semi-group for which the functions $\| T ( t) x \|$ are summable on any finite interval for all $x \in X$. A summable semi-group has a generating operator $A = \overline{ {A _ {0} }}\;$. The operator $A _ {0}$ is closed if and only if, for every $x \in X$,

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

For $\mathop{\rm Re} \lambda > \omega$ one can define the Laplace transform of a summable semi-group,

$$\tag{2 } \int\limits _ { 0 } ^ \infty e ^ {- \lambda t } T ( t) x d t = - R ( \lambda ) x ,$$

giving a bounded linear operator $R ( \lambda )$ which has many properties of a resolvent operator.

A closed operator $A$ with domain of definition dense in $X$ is the generating operator of a summable semi-group $T ( t)$ if and only if, for some $\omega$, the resolvent $R ( \lambda , A )$ exists for $\mathop{\rm Re} \lambda > \omega$ and the following conditions hold: a) $\| R ( \lambda , A ) \| \leq M$, $\mathop{\rm Re} \lambda > \omega$; b) there exist a non-negative function $\phi ( t , x )$, $t > 0$, $x \in X$, jointly continuous in all its variables, and a non-negative function $\phi ( t)$, bounded on any interval $[ a , b ] \subset ( 0 , \infty )$, such that, for $\omega _ {1} > \omega$,

$$\int\limits _ { 0 } ^ \infty e ^ {- \omega _ {1} t } \phi ( t , x ) dt < \infty ,$$

$$\overline{\lim\limits}\; _ {t \rightarrow \infty } t ^ {-} 1 \mathop{\rm ln} \phi ( t) < \infty ,\ \phi ( t , x ) \leq \phi ( t) \| x \| ,$$

$$\| R ^ {n} ( \lambda , A ) x \| \leq \frac{1}{ ( n - 1 ) ! } \int\limits _ { 0 } ^ \infty t ^ {n-} 1 e ^ {- \lambda t } \phi ( t , x ) dt .$$

Under these conditions

$$\| T ( t) x \| \leq \phi ( t , x ) ,\ \ \| T ( t) \| \leq \phi ( t) .$$

If one requires in addition that the function $\| T ( t) \|$ be summable on finite intervals, a necessary and sufficient condition is the existence of a continuous function $\phi ( t)$ such that, for $\omega _ {1} > \omega$,

$$\tag{3 } \int\limits _ { 0 } ^ \infty \phi ( t) e ^ {- \omega _ {1} t } dt < \infty ,$$

$$\tag{4 } \| R ^ {n} ( \lambda , A ) \| \leq \frac{1}{( n - 1 ) ! } \int\limits _ { 0 } ^ \infty t ^ {n-} 1 e ^ {- \lambda t } \phi ( t) dt ,$$

$$\lambda > \omega ,\ n= 1 , 2 , . . . .$$

Under these conditions, $\| T ( t) \| \leq \phi ( t)$. By choosing different functions satisfying (3), one can define different subclasses of summable semi-groups. If $\phi ( t) = Me ^ {\omega t }$, the result is the class $C _ {0}$ and (1) follows from (4). If $\phi ( t) = Mt ^ {- \alpha } e ^ {\omega t }$, $0 \leq \alpha < 1$, condition (4) implies the condition

$$\| R ^ {n} ( \lambda , A ) \| \leq \frac{M \Gamma ( n - \alpha ) }{( n - 1 ) ! ( \lambda - \omega ) ^ {n - \alpha } } ,\ \ \lambda > \omega ,\ n = 1 , 2 ,\dots.$$

## Semi-groups with power singularities.

If in the previous example $\alpha \geq 1$, then the integrals in (4) are divergent for $n \leq \alpha - 1$. Hence the generating operator for the corresponding semi-group may not have a resolvent for any $\lambda$, i.e. it may have a spectrum equal to the entire complex plane. However, for $n$ large enough one can define for such operators functions $S _ {n} ( \lambda , A )$ which coincide with the functions $R ^ {n+} 1 ( \lambda , A )$ in the previous cases. The operator function $S _ {n} ( \lambda , A )$ is called a resolvent of order $n$ if it is analytic in some domain $G \subset \mathbf C$ and if for $\lambda \in G$,

$$S _ {n} ( \lambda , A ) Ax = A S _ {n} ( \lambda , A ) x ,\ \ x \in D ( A) ,$$

$$S _ {n} ( \lambda , A ) ( A - \lambda ) ^ {n+} 1 x = x ,\ x \in D ( A ^ {n+} 1 ) ,$$

and if $S _ {n} ( \lambda , A ) x = 0$ for all $\lambda \in G$ implies $x = 0$. If $\overline{D}\; ( A ^ {n+} 1 ) = X$, the operator may have a unique resolvent of order $n$, for which there is a maximal domain of analyticity, known as the resolvent set of order $n$.

Let $T ( t)$ be a strongly-continuous semi-group such that the inequality

$$\| T ( t) \| \leq M t ^ {- \alpha } e ^ {\omega t }$$

holds for $\alpha \geq 1$. Then its generating operator $B$ has a resolvent of order $n$ for $n > \alpha - 1$, and, moreover,

$$S _ {n} ( \lambda , B ) x = \ \frac{1}{n!} \int\limits _ { 0 } ^ \infty t ^ {n} e ^ {- \lambda t } T ( t) x dt ,\ \mathop{\rm Re} \lambda > \omega ,$$

$$\tag{5 } \left \| \frac{d ^ {k} S _ {n} ( \lambda , B ) }{d \lambda ^ {k} } x \right \| \leq \frac{M \Gamma ( k + n + 1 - \alpha ) }{n ! ( \mathop{\rm Re} \lambda - \omega ) ^ {k + n + 1 - \alpha } } ,$$

$$\mathop{\rm Re} \lambda > \omega ,\ k = 0 , 1 ,\dots .$$

Conversely, suppose that for $\mathop{\rm Re} \lambda > 0$ the operator $B$ has a resolvent $S _ {n} ( \lambda , B )$ of order $n$ satisfying (5) with $n > \alpha - 1$. Then there exists a unique semi-group $T ( t)$ such that

$$\| T ( t) \| \leq M t ^ {- \alpha } e ^ {\omega t } ,$$

and the generating operator $A$ of this semi-group is such that $S _ {n} ( \lambda , A ) = S _ {n} ( \lambda , B )$.

## Smooth semi-groups.

If $x \in D ( A _ {0} )$, the function $T ( t) x$ is continuously differentiable and

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

There exist semi-groups of class $C _ {0}$ such that, if $x \notin D ( A _ {0} ) = D ( A)$, the functions $T ( t) x$ are non-differentiable for all $t$. However, there are important classes of semi-groups for which the degree of smoothness increases with increasing $t$. If the functions $T ( t) x$, $t > t _ {0}$, are differentiable for any $x \in X$, then it follows from the semi-group property that the $T ( t) x$ are twice differentiable if $t > 2 t _ {0}$, three times differentiable if $t > 3 t _ {0}$, etc. Therefore, if these functions are differentiable at any $t > 0$ for $x \in X$, then $T ( t) x$ is infinitely differentiable.

Given a semi-group of class $C _ {0}$, a necessary and sufficient condition for the functions $T ( t) x$ to be differentiable for all $x \in X$ and $t > t _ {0}$, where $t _ {0} \geq 0$, is that there exist numbers $a , b , c > 0$ such that the resolvent $R ( \lambda , A )$ is defined in the domain

$$\mathop{\rm Re} \lambda > a - b \mathop{\rm ln} | \mathop{\rm Im} \lambda | ,$$

while in this domain

$$\| R( \lambda , A ) \| \leq c | \mathop{\rm Im} \lambda | .$$

A necessary and sufficient condition for $T ( t) x$ to be infinitely differentiable for all $x \in X$ and $t > 0$ is that, for every $b > 0$, there exist $a _ {b} , c _ {b}$ such that the resolvent $R ( \lambda , A )$ is defined in the domain

$$\mathop{\rm Re} \lambda > a _ {b} - b \mathop{\rm ln} | \mathop{\rm Im} \lambda | ,$$

and such that

$$\| R ( \lambda , A ) \| \leq c _ {b} | \mathop{\rm Im} \lambda | .$$

Sufficient conditions are: If there exists a $\mu > \omega$ for which

$$\overline{\lim\limits}\; _ {\tau \rightarrow \infty } \mathop{\rm ln} | \tau | \| R ( \mu + i \tau , A ) \| = t _ {0} < \infty ,$$

then the $T ( t) x$ are differentiable for $t > t _ {0}$ and $x \in X$; if $t _ {0} = 0$, then the $T ( t) x$ are infinitely differentiable for all $t > 0$ and $x \in X$.

The degree of smoothness of a semi-group may sometimes be inferred from its behaviour at zero; for example, suppose that for every $c > 0$ there exists a $\delta _ {c}$ such that, for $0 < t < \delta _ {c}$,

$$\| I - T ( t) \| \leq 2 - ct \mathop{\rm ln} t ^ {-} 1 ,$$

then the $T ( t) x$ are infinitely differentiable for all $t > 0$, $x \in X$.

There are smoothness conditions for summable semi-groups and semi-groups of polynomial growth. If a semi-group has polynomial growth of degree $\alpha$ and is infinitely differentiable for $t > 0$, then the function

$$\frac{d T ( t) }{dt} x = A T ( t) x$$

also has polynomial growth:

$$\| A T ( t) \| \leq M _ {1} t ^ {- \beta } e ^ {\omega t } .$$

In the general case there is no rigorous relationship between the numbers $\alpha$ and $\beta$, and $\beta$ can be utilized for a more detailed classification of infinitely-differentiable semi-groups of polynomial growth.

## Analytic semi-groups.

An important class of semi-groups, related to partial differential equations of parabolic type, comprises those semi-groups $T ( t)$ which admit an analytic continuation to some sector of the complex plane containing the positive real axis. A semi-group of class $C _ {0}$ has this property if and only if its resolvent satisfies the following inequality in some right half-plane $\mathop{\rm Re} \lambda > \omega$:

$$\| R ( \lambda , A ) \| \leq M | \lambda - \omega | ^ {-} 1 .$$

Another necessary and sufficient conditions is: The semi-group is strongly differentiable and its derivative satisfies the estimate

$$\left \| \frac{d T }{dt} ( t) \right \| \leq M t ^ {-} 1 e ^ {\omega t } .$$

Finally, the inequality

$$\overline{\lim\limits}\; _ {t \rightarrow 0 } \| I - T ( t) \| < 2$$

is also a sufficient condition for $T ( t)$ to be analytic.

If a semi-group $T ( t)$ has an analytic continuation $T ( z)$ to a sector $| \mathop{\rm arg} z | < \phi \leq \pi / 2$ and has polynomial growth at zero, $\| T ( z) \| \leq c | z | ^ \alpha$, $\alpha > 0$, then the resolvent $S _ {n} ( \lambda , A )$ of order $n > \alpha - 1$ of its generating operator $A$ has an analytic continuation to the sector $| \mathop{\rm arg} \lambda | \leq \pi / 2 + \phi$, and satisfies the following estimate in any sector $| \mathop{\rm arg} \lambda | \leq \pi / 2 + \psi$, $\psi < \phi$:

$$\| S _ {n} ( \lambda , A ) \| \leq | \lambda | ^ {\alpha - n - 1 } M ( \psi ) .$$

Conversely, suppose that the resolvent $S _ {n} ( \lambda , B )$ of an operator $B$ is defined in a sector $| \mathop{\rm arg} \lambda | \leq \pi / 2 + \psi$ and that

$$\| S _ {n} ( \lambda , B ) \| \leq \lambda ^ {\alpha - n - 1 } M .$$

Then there exists a semi-group $T ( z)$ of growth $\alpha$, analytic in the sector $| \mathop{\rm arg} z | < \psi$, whose generating operator $A$ is such that $S _ {n} ( \lambda , A ) = S _ {n} ( \lambda , B )$.

## Distribution semi-groups.

In accordance with the general concept of the theory of distributions (cf. Generalized function), one can drop the requirement that the operator-valued function $T ( t)$ be defined for every $t > 0$, demanding only that it be possible to evaluate the integrals $\int _ {- \infty } ^ {+ \infty } T ( t) \phi ( t) dt$ for all $\phi$ in the space $D ( \mathbf R )$ of infinitely-differentiable functions with compact support. Hence the following definition: A distribution semi-group on a Banach space $X$ is a continuous linear mapping $T ( \phi )$ of $D ( \mathbf R )$ into the space $L ( X)$ of all bounded linear operators on $X$, with the following properties: a) $T ( \phi ) = 0$ if $\supp \phi \subset ( - \infty , 0 )$; b) if $\phi , \psi$ are functions in the subspace $D ^ {+} ( \mathbf R )$ of all functions in $D ( \mathbf R )$ with support in $( 0 , \infty )$, then $T ( \phi * \psi ) = T ( \phi ) T ( \psi )$, where the star denotes convolution:

$$\phi * \psi = \int\limits _ {- \infty } ^ \infty \phi ( t - s ) \psi ( s) d s$$

(the semi-group property); c) if $T ( \phi ) x = 0$ for all $\phi \in D ^ {+} ( \mathbf R )$, then $x = 0$; d) the linear hull of the set of all values of $T ( \phi ) x$, $\phi \in D ^ {+} ( \mathbf R )$, $x \in X$, is dense in $X$; e) for any $y = T ( \psi ) x$, $\psi \in D ^ {+} ( \mathbf R )$, there exists a continuous $u ( t)$ on $( 0 , \infty )$ with values in $X$, so that $u( 0) = y$ and

$$T ( \phi ) y = \int\limits _ { 0 } ^ \infty \phi ( t) u ( t) dt$$

for all $\phi \in D ( \mathbf R )$.

The infinitesimal operator $A _ {0}$ of a distribution semi-group is defined as follows. If there exists a delta-sequence $\{ \rho _ {n} \} \subset D ^ {+} ( \mathbf R )$ such that $T ( \rho _ {n} ) x \rightarrow x$ and $T ( - \rho _ {n} ^ \prime ) x \rightarrow y$ as $n \rightarrow \infty$, then $x \in D ( A _ {0} )$ and $y = A _ {0} x$. The infinitesimal operator has a closure $A = \overline{ {A _ {0} }}\;$, known as the infinitesimal generator of the distribution semi-group. The set $\cap _ {n=} 1 ^ \infty D ( A _ {0} ^ {n} )$ is dense in $X$ and contains $T ( \phi ) X$ for any $\phi \in D ^ {+} ( \mathbf R )$.

A closed linear operator $A$ with a dense domain of definition in $X$ is the infinitesimal generator of a distribution semi-group if and only if there exist numbers $a , b \geq 0$, $c > 0$ and a natural number $m$ such that the resolvent $R ( \lambda , A )$ exists for $\mathop{\rm Re} \lambda \geq a \mathop{\rm ln} ( 1 + | \lambda | ) + b$ and satisfies the inequality

$$\tag{6 } \| R ( \lambda , A ) \| \leq c ( 1 + | \lambda | ) ^ {m} .$$

If $A$ is a closed linear operator on $X$, then the set $\cap _ {n=} 1 ^ \infty D ( A ^ {n} )$ can be made into a Fréchet space $X _ \infty$ by introducing the system of norms

$$\| x \| _ {n} = \ \sum _ { k= } 0 ^ { n } \| A ^ {k} x \| .$$

The restriction $A _ \infty$ of $A$ to $X _ \infty$ leaves $X _ \infty$ invariant. If $A$ is the infinitesimal generator of a semi-group, then $A _ \infty$ is the infinitesimal generator of a semi-group of class $C _ {0}$( continuous for $t \geq 0$, $T ( 0 ) = I$) on $X _ \infty$. Conversely, if $X _ \infty$ is dense in $X$, the operator $A$ has a non-empty resolvent set and $A$ is the infinitesimal generator of a semi-group of class $C _ {0}$ on $X _ \infty$, then $A$ is the infinitesimal generator of a distribution semi-group on $X$.

A distribution semi-group has exponential growth of order at most $q$, $1 \leq q < \infty$, if there exists an $\omega > 0$ such that $\mathop{\rm exp} ( - \omega t ^ {q} ) T ( \phi )$ is a continuous mapping in the topology induced on $D ^ {+}$ by the space $S ( \mathbf R )$ of rapidly-decreasing functions. A closed linear operator is the infinitesimal generator of a distribution semi-group with the above property if and only if it has a resolvent $R ( \lambda , A )$ which satisfies (6) in the domain

$$\{ \lambda : { \mathop{\rm Re} \lambda \geq [ \alpha \mathop{\rm ln} ( 1 + | \mathop{\rm Im} \ \lambda | + \beta ) ] ^ {1- 1/q } , \mathop{\rm Re} \lambda > \omega } \} ,$$

where $\alpha , \beta > 0$. In particular, if $q = 1$ the semi-group is said to be exponential and inequality (6) is valid in some half-plane. There exists a characterization of the semi-groups of the above types in terms of the operator $A _ \infty$. Questions of smoothness and analyticity have also been investigated for distribution semi-groups.

## Semi-groups of operators in a (separable) locally convex space $X$.

The definition of a strongly-continuous semi-group of operators $T ( t )$ continuous on $X$ remains the same as for a Banach space. Similarly, the class $C _ {0}$ is defined by the property $T ( t ) x \rightarrow x$ as $t \rightarrow 0$ for any $x \in X$. A semi-group is said to be locally equicontinuous (of class $lC _ {0}$) if the family of operators $T ( t )$ is equicontinuous when $t$ ranges over any finite interval in $( 0 , \infty )$. In a barrelled space, a semi-group of class $C _ {0}$ is always equicontinuous (cf. Equicontinuity).

A semi-group is said to be equicontinuous (of class $uC _ {0}$) if the family $T ( t )$, $0 \leq t < \infty$, is equicontinuous.

Infinitesimal operators and infinitesimal generators are defined as in the Banach space case.

Assume from now on that the space $X$ is sequentially complete. The infinitesimal generator $A$ of a semi-group of class $l C _ {0}$ is identical to the infinitesimal operator; its domain of definition, $D ( A)$, is dense in $X$ and, moreover, the set $\cap _ {n=} 1 ^ \infty D ( A ^ {n} )$ is dense in $X$. The semi-group $T ( t )$ leaves $D ( A)$ invariant and

$$\frac{dT}{dt} ( t) x = \ AT ( t ) x = T ( t ) Ax ,\ 0 \leq t < \infty ,\ x \in D ( A ) .$$

If $A$ is the infinitesimal generator of a semi-group of class $u C _ {0}$, the resolvent $R ( \lambda , A )$ is defined for $\mathop{\rm Re} \lambda > 0$ and is the Laplace transform of the semi-group.

A linear operator $A$ is the infinitesimal generator of a semi-group of class $u C _ {0}$ if and only if it is closed, has dense domain of definition in $X$, and if there exists a sequence of positive numbers $\lambda _ {k} \rightarrow \infty$ such that, for any $\lambda _ {k}$, the resolvent $R ( \lambda _ {k} , A )$ is defined and the family of operators $[ \lambda _ {k} R ( \lambda _ {k} , A ) ] ^ {n}$, $k , n = 1 , 2 \dots$ is equicontinuous. In this situation the semi-group can be constructed by the formula

$$( t ) x = \lim\limits _ {k \rightarrow \infty } \ \left ( \mathop{\rm exp} \left [ - \lambda _ {k} - \lambda _ {k} ^ {2} R ( \lambda _ {k} , A ) \right ] t \right ) x ,$$

$$t \geq 0 ,\ x \in X .$$

In a non-normed locally convex space, the infinitesimal generator of a semi-group of class $lC _ {0}$ may have no resolvent at any point. An example is: $A = d / ds$ in the space $C ^ \infty$ of infinitely-differentiable functions of $s$ on $\mathbf R$. As a substitute for the resolvent one can take a continuous operator whose product with $A - \lambda I$, from the right and the left, differs by a "small amount" from the identity operator.

A continuous operator $R ( \lambda )$ defined for $\lambda$ in a set $\Lambda \subset \mathbf C$ is called an asymptotic resolvent for a linear operator $A$ if $AR ( \lambda )$ is continuous on $X$, the operator $R ( \lambda ) A$ can be extended from $D ( A)$ to a continuous operator $B ( \lambda )$ on $X$, and if there exists a limit point $\lambda _ {0}$ of the set $\Lambda$ such that $H ^ {+} ( \lambda ) x \rightarrow 0$, $H ^ {-} ( \lambda ) x \rightarrow 0$ as $\lambda \rightarrow \lambda _ {0}$ for any $x \in X$, where

$$H ^ {+} ( \lambda ) = ( A - \lambda I ) R ( \lambda ) - I ,\ \ H ^ {-} ( \lambda ) = B ( \lambda ) - \lambda R ( \lambda ) - I .$$

An asymptotic resolvent possesses various properties resembling those of the ordinary resolvent.

A closed linear operator $A$ with a dense domain of definition in $X$ is the infinitesimal generator of a semi-group of class $l C _ {0}$ if and only if there exist numbers $\omega$ and $\alpha > 0$ such that, for $\lambda > \omega$, there exists an asymptotic resolvent $R ( \lambda )$ of $A$ with the properties: the functions $R ( \lambda )$, $H ^ {+} ( \lambda )$, $H ^ {-} ( \lambda )$ are strongly infinitely differentiable for $\lambda > \omega$, and the families of operators

$$e ^ {\alpha \lambda } \frac{d ^ {n} H ^ \pm ( \lambda ) }{d \lambda ^ {n} } ,\ \frac{\lambda ^ {n+} 1 }{n ! } \frac{d ^ {n} R ( \lambda ) }{d \lambda ^ {n} } ,\ \lambda > \omega ,\ n = 0 , 1 \dots$$

are equicontinuous.

Generation theorems have also been proved for other classes of semi-groups of operators on a locally convex space.

If $T ( t )$ is a semi-group of class $C _ {0}$ on a Banach space $X$, then the adjoint operators form a semi-group of bounded operators on the adjoint space $X ^ \prime$. However, the assertion that $T ^ \prime ( t ) f \rightarrow f$ as $t \rightarrow 0$ for any $f \in X ^ \prime$ is valid only in the sense of the weak- topology $\sigma ( X ^ \prime , X )$. If $A$ is the generating operator, its adjoint $A ^ \prime$ is a weak infinitesimal generator for $T ^ \prime ( t )$, in the sense that $D ( A ^ \prime )$ is the set of all $f$ for which the limit of $t ^ {-} 1 [ T ^ \prime ( t ) - I ] f$ as $t\rightarrow 0$ exists in the sense of weak- convergence and is equal to $A ^ \prime f$. The domain of definition $D ( A ^ \prime )$ is dense in $X ^ \prime$— again in the sense of the weak- topology — and the operator $A ^ \prime$ is closed in the weak- topology.

Let $X ^ {+}$ be the set of all elements in $X ^ \prime$ such that $T ^ \prime ( f ) \rightarrow f$ as $t \rightarrow 0$ in the strong sense; then $X ^ {+}$ is a closed subspace of $X ^ \prime$ that is invariant under all $T ^ \prime ( t )$. On $X ^ {+}$ the operators $T ^ \prime ( t )$ form a semi-group of class $C _ {0}$. The space $X ^ {+}$ is also the strong closure of the set $D ( A ^ \prime )$ in $X ^ \prime$. If the original space is reflexive, then $X ^ {+} = X ^ \prime$. Analogous propositions hold for semi-groups of class $C _ {0}$ in locally convex spaces. Semi-groups of classes $l C _ {0}$ and $u C _ {0}$ generate semi-groups of the same classes in $X ^ {+}$.

## Distribution semi-groups in a (separable) locally convex space.

A distribution semi-group $T$ in a sequentially complete locally convex space is defined just as in a Banach space. A semi-group $T$ is said to be locally equicontinuous (of class $lD ^ \prime$) if, for any compact subset $K \subset D ( \mathbf R )$, the family of operators $\{ T ( \phi ) \}$, $\phi \in K$, is equicontinuous. In a barrelled space $X$, any distribution semi-group is defined by analogy to the Banach case. For semi-groups of class $l D ^ \prime$, the infinitesimal operator is closed $( A _ {0} = A )$, $\cap _ {n=} 1 ^ \infty D ( A ^ {n} )$ is dense in $X$, and for any $x \in X$ and $\phi \in D ( \mathbf R )$,

$$\tag{7 } \left . \begin{array}{c} {T ( \phi ) x \in D ( A ) ,\ T ^ \prime ( \phi ) x = AT ( \phi ) x + \phi ( 0 ) x , } \\ {T ^ \prime ( \phi ) x = T ( \phi ) Ax + \phi ( 0 ) x ,\ x \in D ( A ). } \end{array} \right \}$$

A generalized function $T$ with support in $[ 0 , \infty )$, possessing the properties (7), is naturally called the fundamental function of the operator $( d / dt ) - A$. Thus, if $A$ is the infinitesimal operator of a semi-group $T$ of class $l D ^ \prime$, then $T$ is the fundamental function of the operator $( d / dt ) - A$. The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function $T$( or, more precisely, of the function $f ( T ( \phi ) x )$, where $f \in X ^ \prime$).

A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let $\widehat \phi$ denote the Laplace transform of a function $\phi \in D ( \mathbf R )$, and let $\widehat{D} ( \mathbf R )$ be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of $D ( \mathbf R )$. The Laplace transform of an $X$- valued generalized function $F$ is defined by $\widehat{F} ( \widehat \phi ) = F ( \phi )$. Under these conditions, $\widehat{F}$ is a continuous mapping of $\widehat{D} ( \mathbf R )$ into the space $L ( X )$ of continuous linear operators on $X$. Let $\widehat{D} {} _ {+} ^ \prime$ be the space of all $\widehat{F}$ obtained from functions $F$ with support in $( 0 , \infty )$, with the natural topology. If $A$ is a linear operator on $X$, it can be "lifted" to an operator $\widetilde{A}$ on $\widehat{D} {} _ {+} ^ \prime$ via the equality

$$( \widetilde{A} \widehat{F} ) = A ( \widehat{F} ( \widehat \phi ) ) = A F ( \phi ) .$$

Thus, it is defined for all $\widehat{F} \in \widehat{D} {} _ {+} ^ \prime$ such that the right-hand side of the equality is defined for any $\phi \in D ( \mathbf R )$ and it extends to a generalized function in $\widehat{D} {} _ {+} ^ \prime$. The continuous operator $\widetilde \lambda$ on $\widehat{D} {} _ {+} ^ \prime$ is defined by

$$( \widetilde \lambda \widehat{F} ) ( \widehat \phi ) = \lambda \widehat{F} ( \widehat \phi ) = F ^ \prime ( \phi ) = - F ( \phi ^ \prime ) .$$

If the operator $\widetilde{A} - \widetilde \lambda$ has a continuous inverse $\widetilde{R}$ on $\widehat{D} {} _ {+} ^ \prime$, then $\widetilde{R}$ is called the generalized resolvent of $A$.

An operator $A$ has a generalized resolvent if and only if the operator $( d / dt ) - A$ has a locally equicontinuous fundamental function $T$, constructed by the formula

$$T ( \phi ) x = ( \widetilde{R} ( 1 \otimes x ) ) ( \widehat \phi ) ,\ \phi \in D ( \mathbf R ) ,\ \ x \in X ,$$

where

$$( 1 \otimes x ) ( \widehat \phi ) = ( \delta \otimes x ) \phi = \phi ( 0 ) x .$$

Subject to certain additional assumptions, $T$ is a distribution semi-group. An extension theorem for semi-groups of class $l C _ {0}$ has also been proved in terms of generalized resolvents.