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I have a little bit confused on unitary operators and surjective isometries on a Hilbert space.

I think it is quite clear that A operator is unitary if and only if it is a surjective isometry.

However, according to the first line of the 2nd page of https://link.springer.com/article/10.1007/BF02761592 (or first paragraph of the 2nd page of https://core.ac.uk/download/pdf/82282502.pdf), there are more surjective isometries on a Hilbert space onto itself than unitary operators.

Is it a contradiction or did I misunderstand something?

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  • $\begingroup$ I think the paper is just saying that there are actually many more isometries in a Hilbert space than just those that conform to the form of (1.2) and (1.3). Indeed, the author states that these other isometries can’t be composed out of unitary operators as (1.2) or (1.3) would suggest, so the isometries the author is referring to are probably in particular those non-surjective ones. You are correct in thinking that all surjective isometries are unitary operators and vice-versa. $\endgroup$ May 20, 2019 at 2:09
  • $\begingroup$ @JackCrawford Actually, these authors considered surjective/onto isometries...They emphasized "onto" and “surjective” in their papers. $\endgroup$
    – user92646
    May 20, 2019 at 2:14
  • $\begingroup$ I see them refer to “isometries of a set onto itself” but I don’t think the term “onto” here is to be read as surjectivity so much as just specifying the codomain. I’m afraid I can’t spot any location where they refer to these isometries (that explicitly do not conform to (1.2) and (1.3)) as surjective, sorry, could you point out a location where the authors say this? $\endgroup$ May 20, 2019 at 2:18
  • $\begingroup$ @JackCrawford Please don't be sorry. I am very grateful for your help. In the first paper, the author didn't say surjective but if u and w are unitary operator, then the isometry is clearly surjective. Well, I get your point. You mean in p248, the author didn't say that the isometry is surjective. So, please have a look at the second paper. In line 8 of the second page, it is explicitly pointed out that the isometries are surjective. $\endgroup$
    – user92646
    May 20, 2019 at 2:23
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    $\begingroup$ @JackCrawford I reckon that the first author made a mistake and then the second just followed... $\endgroup$
    – user92646
    May 20, 2019 at 2:40

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You seem to be confusing what the authors are saying. They do not state that there are more surjective isometries on a Hilbert space than unitaries.

In the quote from Arazy's paper, the author states that isometries of $C_2$ (i.e. the Hilbert-Schmidt operators) onto itself, of the form $x\mapsto uxw$ for some unitaries $u,w\in\mathcal B(\ell^2)$ does not exhaust the set of all surjective (linear) isometries of $C_2$ to itself, but that it does for $C_p$ with $p\neq 2$ (i.e. the set of Schatten $p$-class operators).

In the quote from Sourour's paper, he states that for a Hilbert space $H$, maps from $\mathcal B(H)\to\mathcal B(H)$ of the form $x\mapsto uxw$ for unitaries $u,w\in\mathcal B(H)$, exhaust the set of all surjective (linear) isometries of $\mathcal B(H)$ to itself.

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  • $\begingroup$ Ah, the reason is there are two Hilbert spaces $C_2$ and $\ell_2$ here! Thanks! $\endgroup$
    – user92646
    May 20, 2019 at 5:37
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    $\begingroup$ You're welcome. Glad to help! $\endgroup$
    – Aweygan
    May 20, 2019 at 6:24

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