infinite direct sum of isometries

Question

Suppose $H, K$ are two Hilbert spaces, let $\oplus K$ be the inifinite direct sum of $K$, if there exists an isometry $V$ from $\oplus K$ to $H$. In Davidson's book, the author mentions that we can write $V$ as the form of $(V_1,\cdots,V_n,\cdots).$, where each $V_i$ is an isomery. How to define each $V_i$?

Answer 1

Assuming that your direct sum is a countably infinite direct sum $\oplus K_i$, $V_n$ is simply the restriction of $V$ to $K_n$.