## Semigroups

The semigroup theory for operators was inspired by the exponential of a bounded operator $A \in \mathcal{B}(X)$, namely $e^{A}=\sum_{k=0}^\infty A^k/(k!)$, series which is absolutely convergent for $A$ bounded and therefore convergent. The semigroup theory is an adaptation of this for unbounded operators $A$. The purpose of this page is to present some interesting properties and theorems concerning these families of operators, which will allow me to find some interesting counterexamples in operator theory (for example a closed operator densely defined with void spectrum, with unbounded spectrum, etc.).

Proofs and extensions of the presented facts can be found in the following works:

Short Course on Operator Semigroups, K.J Engel, R. Nagel

One Parameter Semigroups for Linear Evolution Equations, K.J. Engel, R. Nagel

Semigroups of Linear Operators and Applications to Partial Differential Equations, A. Pazy

In further considerations unless stated otherwise $X$ will denote a Banach space.

Definition: A family of operators $T(t): [0,\infty) \to \mathcal{B}(X)$ is called a $\mathcal{C}_0$-semigroup if it satisfies the following conditions:

1. $T(0)=I$;
2. $T(s+t)=T(s)T(t),\ \forall s,t \in [0,\infty)$;
3. $\lim\limits_{t \to 0_+} T(t)x=x,\ \forall x \in X$.

One of its properties is that $\|T(t)\|$ does not grow faster than an exponential.

Exponential Growth Theorem: Let $\{T_t\}_{t \geq 0}$ be a $\mathcal{C}_0$-semigroup. Then there exists $M \geq 1$ and $\omega \geq 0$ such that $\|T(t)\|\leq Me^{\omega t},\ \forall t \geq 0$.

Exponential Growth Index: The last theorem allows us to define $\omega_0(T)= \inf \{\omega \in \Bbb{R} : \exists M>0 \text{ such that }\|T(t)\|\leq Me^{\omega t},\ \forall t \geq 0\}$.

We can prove the following property:
$\omega_0(T)=\displaystyle\inf_{t >0} \frac{\ln\|T(t)\|}{t}=\lim_{t \to \infty} \frac{\ln \|T(t)\|}{t}$.

It is true that the application $t \mapsto T(t)x : \Bbb{R}_+ \to X$ is continuous. In the following, we show who plays the role of the bounded operator in the definition of $e^{A}$ in the case of semigroups.

Infinitesimal Generator: Define
$\displaystyle D(A):=\{x \in X : \exists \lim_{h \to 0_+} \frac{T(h)x-x}{h} \}$ and
$\displaystyle A:D(A) \to X,\ Ax=\lim_{h \to 0_+} \frac{T(h)x-x}{h}$.

Properties of the generator:
If $x \in D(A)$ then

1. $T(t)x \in D(A),\ \forall t \geq 0$ and $AT(t)x=T(t)Ax$;
2. The mapping $T(\cdot)x : \Bbb{R}_+ \to X$ is differentiable and $\displaystyle \frac{d}{dt}T(t)x=AT(t)x=T(t)Ax$;
3. $T(t)x-T(s)x=\displaystyle \int_s^t T(\tau)Ax d\tau$.
4. $\displaystyle \int_0^t T(\tau)x d \tau \in D(A),\ \forall x \in X,\ \forall t \geq 0$, and $\displaystyle A\int_0^t T(\tau)x d \tau=T(t)x-x$;
5. $\overline{D(A)}=X$, namely $A$ is densely defined;
6. Two semigroups having the same generator are equal. This shows us it makes sense to name this operator the generator of our semigroup.
7. The generator is a closed operator.
8. If $\lambda \in C$ and $Re \lambda > \omega_0(T)$ then $\lambda \in \rho(A)$ and $R(\lambda;A)x=\displaystyle \int_0^\infty e^{-\lambda t}T(t)x dt$.

If one of the operators $T(t),\ t>0$ is invertible, then $T(t)$ is invertible for all $t >0$. If this happens, then $\{T_t^{-1}\}$ is also a $\mathcal{C}_0$ semigroup, and if the generator of the initial semigroup is $A$, then the generator of the inverse semigroup is $-A$.