# 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.

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