I have been studying Scattering theory book by Lax-Phillips. In Chapter 1 of the book, they describe scattering theory in terms of representation theory. To explain briefly, say $U(t)$ is a group of operators relating initial data (this is a Hilbert space $H$) to data at time $t$, $D_{-}$ and $D_{+}$ are incoming and outgoing subspaces respectively. $D_{-}( D_{+}) $ consists of initial data of those solutions of wave equation which are zero for all negative (positive) time in some spherical neighborhood of $x$.
The book then goes on to describe 4 conditions that these subspaces satisfy. Moreover, they go on to say that $H$ can be represented isometrically as $L_2((0, \infty), N)$, where $N$ is an auxiliary Hilbert space, and that is equivalent to saying that $U(t)$ acts as translation to the right by $t$ units.
I have 2 questions -
One of the conditions they give is - "$D_{+}(t)$ is decreasing family of subspaces and $D_{-}(t)$ is an increasing family of subspaces." Can anyone explain how this follows from the definition of incoming and outgoing subspace?
Why does translation to right by $t$ units as a translation representation imply $H$ can be isometrically represented as $L_2((0, \infty), N)$?
I am really at a loss. Any help will be appreciated!