Every surjective isometry on a Hilbert space is indeed a unitary operator

Question

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?

Answer 1

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.