An example of a non unitary isometry on $L^2(\mathbb{R})$?

Question

I would like to see one or preferrably two isometries on $L^2(\mathbb{R})$ which are non surjective (equivalently non unitaries)?

Thanks in advance.

Math.

Answer 1

I'll give you uncountably many non-unitary isometries. For $T>0$, define $U_T:L^2(\mathbb R)\to L^2(\mathbb R)$ by \begin{align*} (U_Tf)(t)=\left\{\begin{array}{lcl} f(t-T) &:& t\in[T,\infty),\\ 0 &:& t\in[0,T),\\ f(t)&:&t\in(-\infty,0). \end{array}\right. \end{align*}

That is $$U_Tf (x) = \mathbb{1}_{(-\infty,0)}(x)f(x) + f(x-T)\mathbb{1}_{[T,\infty)}(x)$$