Semigroups
The semigroup theory for operators was inspired by the exponential of a bounded operator , namely , series which is absolutely convergent for bounded and therefore convergent. The semigroup theory is an adaptation of this for unbounded operators . 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 will denote a Banach space.
Definition: A family of operators is called a semigroup if it satisfies the following conditions:
 ;
 ;
 .
One of its properties is that does not grow faster than an exponential.
Exponential Growth Theorem: Let be a semigroup. Then there exists and such that .
Exponential Growth Index: The last theorem allows us to define .
We can prove the following property:
.
It is true that the application is continuous. In the following, we show who plays the role of the bounded operator in the definition of in the case of semigroups.
Infinitesimal Generator: Define
and
.
Properties of the generator:
If then
 and ;
 The mapping is differentiable and ;
 .
 , and ;
 , namely is densely defined;
 Two semigroups having the same generator are equal. This shows us it makes sense to name this operator the generator of our semigroup.
 The generator is a closed operator.
 If and then and .
If one of the operators is invertible, then is invertible for all . If this happens, then is also a semigroup, and if the generator of the initial semigroup is , then the generator of the inverse semigroup is .

January 15, 2011 at 10:43 pmSemigroup Examples « Problems – Beni Bogoşel