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Let $X$ be a Hilbert space and let $(T(t))_{t\geq 0}$ a $C_0$-semigroup on $X$, we recall that:

1- $(T(t))_{t\geq 0}$ is said to be uniformly bounded if there exists $M\geq 0$ such that for all $t\geq 0$ $$\|T(t)\|\leq M,$$ where $\|T(t)\|$ is the operator norm of the bounded operator $T(t)$.

2- $(T(t))$ is said to be strongly stable if for all $x\in X$, $$\lim_{t\rightarrow\infty}T(t)x=0.$$

I wonder if it is possible to find a $C_0$-semigoup on $X$ which is strongly stable but not uniformly bounded.

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The second condition means that $\|T(t)x\|$ $t>0$ is bounded for all $x$. Hence, by the uniform boundedness principle, 1. holds. –  András Bátkai Jun 8 '13 at 18:18
    
Thanks, I think it responds to the question. –  driss-alamilouati Jun 8 '13 at 19:28
    
You are welcome :-) –  András Bátkai Jun 8 '13 at 21:05

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