Isometric operators close to the identity

Question

In a separable Hilbert space $\mathcal{H}$ there are unitary operators in the neighborhood of the identity. For instance, via the Stone's theorem we can construct a strongly continuous one-parameter unitary group $U(t)$ such that $U(0)=\mathbb{I}$. My question is whether is this also true for isometric operators which are not unitary. Namely, can we construct a strongly continuous one-parameter family of isometric operators (but not unitary) $T(t)$ such that $T(0)=\mathbb{I}$?

Answer 1

Of course, take $L^2(\mathbb R_+)$ and the isometries $I_t$ given by $I_t(f)(s) = f(s - t) \mathbb{1}_{[t, \infty)}$