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Here is a simple argument showing that with the above notation, we have $d(\cal{U}(n),\tilde{\cal{U}}(n))\ge \sqrt{2}$. Let $U$ be a unitary operator, and $\tilde U$ an antiunitary operator in $\mathb …

This question is related to the previous post, "A question about unitary and anti-unitary matrices". Following the suggestion of Lspice, I am posting it as a separate question, as it might be of indep …

The question is the following: Let $U:\mathbf{C}^n\to \mathbf{C}^n$ be a unitary operator; let $\tilde{U}:\mathbf{C}^n\to\mathbf{C}^n$ be an antiunitary operator. Can one deform $U$ to $\tilde{U}$ wit …

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Well, it seems that the claim is false if $n$ is even: In that case both $U$ and $\tilde{U}$ belong to $\operatorname{SO}(2n)$, w …

$\newcommand\norm[1]{\lVert#1\rVert}$Here is a simple argument showing that, with the above notations, if $U$ is a unitary operator, and $\tilde{U}$ an antiunitary operator in $\mathbf{C}^n$, one alwa …