Why do time evolution semigroups have to be contracting?

Question

Studying the theory of open quantum systems, some textbooks start by introducing the notion of a semigroup for the time evolution operator $T_t$. The next step then usually is to impose that such a semigroup should be contracting, i.e. $||T_t|| leq 1$ for every $tgeq0$. This can be found for example in Open Quantum Systems: An Introduction by Rivas & Huelga.
Why is it important physically, that the time evolution semigroup is contracting? Why do we not demand that it conserves the norm of a state, i.e. $||T_t|| = 1$?
Update after discussion with Valter Moretti: Is a contracting semigroup always associated to Markovian dynamics? Do all Markovian systems show contracting semigroup time evolutions?

Valter Moretti · Answer

I do not know the general context, but I see that the book deals with open systems. The probability is not conserved, in general, for open systems. However it cannot increase (it remains a probability).
The condition $||T_t||leq 1$ implies $$||T_tpsi|| leq ||T_t||: ||psi_1|| leq 1 cdot 1 =1$$ as wanted, if $psi$ is the normalized initial state vector. On the other hand, if
$$||T_tpsi||leq 1$$ for every normalized vector $psi$, the very definition of norm of operator implies $||T_t|| leq 1$.
In summary, $||T_t||leq 1$ is equivalent to the fact that the probability cannot increase but it may decrease.

Why do time evolution semigroups have to be contracting?

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