Strongly continuous semigroups that cannot be contractions

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.

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 $$\|T(t)f_t\|_{\mu} = 2.$$