ContractiveSemigroup

A semigroup $T(t)$ is said to be a contractive semigroup if it has the following properties:

1. Positivity: For every $t \geq 0$, $T(t)$ is a positive operator, i.e., if $x \geq 0$, then $T(t)x \geq 0$.

2. Strong continuity: The map

$$t \mapsto T(t)x$$

is strongly continuous for every $x$ in the space, i.e., the function is continuous when the space is equipped with the topology derived from its norm.

3. Contractiveness: $T(t)$ is contractive, i.e., for every $x$ and $y$ in the space, the inequality

$$|T(t)x - T(t)y| \leq |x - y|$$ holds.

This property basically says that the operation $T(t)$ doesn't "stretch" the space, but could potentially "shrink" it. This gives us a nice property of convergence in the limit.

A semigroup $T(t)$ is said to be a quasicontractive semigroup if it satisfies all the above properties except the contractiveness. Instead of contractiveness, it satisfies:

1. Quasi-contractiveness: $T(t)$ is quasi-contractive, i.e., there exists a constant $M \geq 1$ such that for every $x$ and $y$ in the space, the inequality

$$|T(t)x - T(t)y| \leq M|x - y|$$ holds.

The key difference from the contractive case is that we now have a factor $M$ that could potentially stretch the space, but only by a bounded amount.