Let $\{T(t),t\ge 0\}$ be a $C_0$ semigroup on a Hilbert space $X$, does that exist a larger Hilbert space $Y$ such that $X\subset Y$, and $T(t)$ extend to a $C_0$ group $T'(t)$ (so $t<0$ make sense now) on $Y$?

**Edit:** Here "extend" means that $(T(t)x, y)=(T'(t)x, y)$ for all $x, y\in X$, which is weaker than its usual meaning. If the semigroup $T(t)$ is a contraction, then the conclusion is true, see Theorem 8.1 (p.29) in "Harmonic analysis of operators on Hilbert spaces" written by Sz. Nagy, C. Foias, H. Bercovici and L. Kérchy.

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