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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.

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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 in

Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.

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, Proc. Am. Math. Soc. 21, 240-244 (1969). ZBL0175.13802.

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    $\begingroup$ Just to point out that the modification of the norm as described for example by Engel and Nagel and quite common in the theory of dynamical systems and facilitates substantially some of the arguments. On refers to them usually as Lyapunov norms and are important for example in smooth ergodic theory. The comments made by András apply verbatim to the construction in dynamics. $\endgroup$
    – John B
    Commented Dec 20, 2015 at 23:37

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