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.


1 Answer 1


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.

Glancing over Douglas, Muhly and Pearcy's paper I would say that invariant subspaces was something they were thinking about and so the co-isometric extension made sense to them. Not to mention the usual bias for upper triangular versus lower triangular. You should also realize that this paper came out a year or two after the Sz. Nagy-Foias and Sarason papers, meaning that there really was no established convention.

One could say that this is also equivalent to whether you prefer to do row operations or column operations on your matrices.

Ultimately you shouldn't think of this as an arbitrary choice. It's like going around a city block, either you go right and then left or left and then right, it often just depends on what you want to see along the way.

  • $\begingroup$ I noted that aswell, hence "There must be a reason why he choose to go to the adjoint...." I dont see why he does it tho and that's what I am looking for. Both Nagy och Foias and Sarason seems to be on the same page while he goes another way. I am intersted since I am trying to understand the modern formulations of this theory which seems to be a blend of a lot of different peoples contributions. This in turn make some ideas look abitrary, which they arent. $\endgroup$
    – user123124
    May 23, 2017 at 10:21

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