# Why do Douglas-Muhly-Pearcy consider the following operator a co-isometry?

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.

An operator $A$ is a co-isometry if its adjoint $A^*$ is an isometry, that is $AA^* = I$. It really is a matter of taste whether you do dilation theory with isometries or co-isometries, in the end you really need a mix of the two to get to the unitary dilation. One consideration is that the minimal co-isometric extension $S$ of a contraction $A \in B(\mathcal H)$ has $\mathcal H$ as an invariant subspace while the minimal isometric co-extension has $\mathcal H$ as a co-invariant subspace.