Semigroup Methods

Chapter 1

Semigroup Methods

1.1 C

-semigroups

We begin with a brief summary of very basic notions and results from the

theory of C

-semigroups.

1.1.1 Denitions and main properties

Let X be a Banach space.

Denition 1.1.1 A one-parameter family of bounded linear operators

fU(t); t 0g is called a C

-semigroup (or semigroup of class C

) if the

following conditions hold

(U1) U(t+ h) = U(t)U(h); t; h 0;

(U2) U(0) = I;

(U3) U(t) is strongly continuous with respect to t 0.