|dc.description.abstract||This work concerns one of the most important tools to solve well-posed problems in the the-
ory of evolution equations (e.g di usion equation, wave equations, ...) and in the theory of
stochastic process, namely the semigroups of linear operators with application to di erential
A semigroup of linear operator on a Banach space X is a continuous operator valued function
T : [0, +∞) → B(X, X) such that T (t + s) = T (t)T (s) for every t, s ≥ 0 and T (0) = I .
The fact that every non zero continuous complex function that satis es f (s+t) = f (s)f (t) for
every t, s ≥ 0 has the form f (t) = exp(at), and that f is determined by the number a = f (0),
T (t)x − x
motivates the association to (T (t)t≥0 ) of an operator A de ned by Ax := lim
x ∈ D(A) and called the in nitesimal generator of (T (t))t≥0 . Furthemore the study of the
converse is of essential interest in the line of Hille-Yosida.
We divide this work into three chapters:
In the rst chapter we present some preliminaries on the spectral theory, most of the ma-
terials follow from A. D. Andrew and W. L. Green; C.E. Chidume , G. Barbatis; E.B.
Davies and J.A. Erdos; Erwin Kreyszig ; Khalil Ezzinbi .
In the second chapter we present the generation and representation of semigroups of linear
operators and provide Hille-Yosida theorem which caracterizes the in nitesimal generator of
a class of Continuous semigroup; essentially most of the materials follow from A. Pazy ;
Khalil Ezzinbi .
Lastly we present the Abstract Cauchy problem as application, essentially most of the ma-
terials follow from Khalil Ezzinbi ; Alain Bensoussen, Guiseppe Da Prato, Michel C.
Delphour, Sanjoy K. Mitter .||en_US