I been reading Nagy and Foias' book "Harmonic analysis of operators on Hilbert space". They prove the existence of a isometric (also unitary) dilation.

However Douglas-Muhly-Pearcy in http://projecteuclid.org/download/pdf_1/euclid.mmj/1029000093 seem to call this a co-isometry. Moreover they refer to Sarason's result on $V^{*}$, that is, an isometry (the forward shift).

Question: Why do they refer to this as a co-isometry? There must be a reason why he choose to go to the adjoint but I do not see it.

I suspect that they consider the backward shift as a more natural contraction or that they are dealing with the unitary dilation (isometry and co-isometry). Either way, it leaves me confused.