Let $H$ be an infinite-dimensional separable Hilbert space. I would like to explicitly construct a family of bounded operators, $(W_t)_{t\in(0,1]}$, with each $W_t:H\rightarrow H$ being an isometry, with the following properties:
- $W_t$ is continuous in the strong operator topology;
- $W_1=id_H$;
- $\lim_{t\rightarrow 0}(W_t W_t^*)=0,$
where the limit is taken in the strong operator topology.
Remark: That it is possible to construct such a family $W_t$ is something I came across in a a paper, although the authors did not give a proof. I see that the infinite-dimensionality of $H$ must be important here, and the right idea seems to be to somehow construct $W_t$ so that the image of $W_t$ as a function of $t$ gets smaller as $t\rightarrow 0$. But it seems to be tricky to explicitly do this and also satisfy the continuity condition.
Identify $H$ with $L^2(\mathbb{R}_+,dm)$, where $dm$ is the Lebesgue measure. Define the isometries $(R_s)_{s \in \mathbb{R}_+}$ by $$ R_s(f)(t) = \begin{cases} f(t - s) & \mbox{ when } t \geq s\\ 0 &\mbox{ otherwise }\end{cases} $$ Then $R_0$ is the identity and $P_s = R_s \, R_s^\ast$ is the projection given by multipliying by $\chi_{[s,\infty)}$, therefore goes to $0$ in the SOT topology as $s \to \infty$. To obtain your $W_t$ just make a continuous change of variable sending $1$ to $0$ and $0$ to $\infty$.