I am currently studying a paper by Jim Agler and Mark Stankus called m-Isometric Transformations of Hilbertspaces which you can find here https://core.ac.uk/download/pdf/19158531.pdf.
My question is how definition 1.8 is to be understood. In the paper it says that
Let $T\in L_b(H)$ be an m-isometry and $l$ a positive integer. T is $l$-pure if T has no nonzero direct summand which is an $l$-isometry.
I tried to decipher this, and my interpretation was
$\forall M\subset H, M \ closed,\ T-reducible,M^\perp \neq {0}:T\restriction_{M^{\perp}}$ is not an $l$-isometry
using this definition it appears to me that on the space $H={0}$ every $m$-isometry is pure. Using some proposition the author further argues in (1.10) that for some $T$-reducible closed subspace $H_m$
$T\restriction_{H_m}$ is a pure m-Isometry (if $H_m\neq${$0$})
I don't get for what purpose he put the "if" there, since the statement (even without knowing anything about $H_m$) surely holds for $H_m=0$.
hence my questions are 1. if my interpretation of the defintion is correct and 2. why he put that "if" there.
thanks in advance