Let $X$ be a Banach space, and $(P\_t)\_{t \ge 0}$ a strongly continuous semigroup of bounded operators on $X$. Using the uniform boundedness principle, it's simple to prove that there are constants $M, \omega$ such that $$\|P\_t\| \le M e^{\omega t} \quad (\*)$$ for all $t$. Moreover, if $M=1$ you can get a contraction semigroup by studying $e^{-\omega t} P\_t$, and I'm having trouble thinking of an example where that's not the case.



> 
> What is a simple example of a semigroup $P\_t$ for which (\*) cannot hold with $M=1$?
> 
> 
> 


If possible, I'd like to see an example where $X$ is a separable Hilbert space.



Strongly continuous semigroups that cannot be contractions
There are many examples constructed with weighted shifts. The following is a Hilbert space example.


Let us consider the Hilbert space $L^2\big((0,1),\mu\big)$, where $\mu$ denotes the measure defined by
$$\mu(A):=2\lambda(A\cap(0,\tfrac12))+\lambda(A\cap(\tfrac12,1)).$$
for all Lebesgue measurable sets A. Here $\lambda$
is the Lebesgue-measure. Furthermore, let $T(t)$ be the nilpotent left shift semigroup. Obviously, $T$ satises the semigroup property and, since the norm
$\|\cdot\|\_{\mu}$ is equivalent to the norm $\|\cdot\|\_{\lambda}$, $T$ is strongly continuous. 


Clearly, $\|T(t)\|\leq 2$.


In addition we see that $T(t) = 0$ for all $t>1$.


Finally, consider the function 
$$f\_t = \frac{1}{\sqrt{t}}\chi\_{(\frac12,\frac12+t)}$$
for $t\in(0,\tfrac12)$.
Clearly, $\|f\_t\|\_{\mu} = 1$ and
$$\|T(t)f\_t\|\_{\mu} = 2.$$


**ADDED:**


Theoretically, you can always introduce an equivalent norm in your space which makes your semigroup a contraction semigroup, see [Lemma II.3.10](https://www.math.uni-tuebingen.de/de/forschung/agfa/members/engel-nagel_one-parameter-semigroups.pdf) in 


*Engel, Klaus-Jochen; Nagel, Rainer*, [**One-parameter semigroups for linear evolution equations**](http://dx.doi.org/10.1007/b97696), Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). [ZBL0952.47036](https://zbmath.org/?q=an:0952.47036).


The construction is, however, in most cases highly non-constructive and can only used for theoretical purposes.


It is more important that there are examples in Hilbert spaces where it is impossible to find an equivalent *Hilbert space* norm (i.e., the semigroup is not similar to a contraction semigroup), see


*Packel, E. W.*, [**A semigroup analogue of Foguel’s counterexample**](http://dx.doi.org/10.2307/2036902), Proc. Am. Math. Soc. 21, 240-244 (1969). [ZBL0175.13802](https://zbmath.org/?q=an:0175.13802). 


