Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm ( of the $2\times 2$ matrices \begin{equation} \Big\{\begin{pmatrix}a & b\\ c& e\end{pmatrix} \Big\vert\ a, b, c, e \in \mathbb{K}, \Big| \Big|{\begin{pmatrix} a & b\\ c& e\end{pmatrix}} \Big| \Big|<1,\ \ \Big| \Big|{\begin{pmatrix} a & \varepsilon b\\ \frac{c}{\varepsilon}& e \end{pmatrix}} \Big| \Big| <1 \Big\} \end{equation} over $\mathbb{K}$ as a function of $\varepsilon \in [0,1]$ is given by \begin{equation} \tilde{\chi}_2 (\varepsilon ) =\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2), \end{equation} when the field $\mathbb{K}$ is the complex one $\mathbb{C}$ and \begin{equation} \tilde{\chi}_4 (\varepsilon ) =\frac{1}{35} \varepsilon^4 (15 \varepsilon^4 -64 \varepsilon^2 +84), \end{equation} when the field $\mathbb{K}$ is the quaternionic one $\mathbb{H}$. (Note that both functions equal 1 at $\varepsilon=1$.)

If the field $\mathbb{K}$ is the real one $\mathbb{R}$, Lovas and Andai ( eq. (9)) have shown that the desired function of $\varepsilon$ is given by \begin{equation} \label{BasicFormula} \tilde{\chi}_1 (\varepsilon ) =1-\frac{4}{\pi^2}\int\limits_\varepsilon^1 \left( s+\frac{1}{s}- \frac{1}{2}\left(s-\frac{1}{s}\right)^2\log \left(\frac{1+s}{1-s}\right) \right)\frac{1}{s} \mbox{d} s \end{equation} \begin{equation} = \frac{4}{\pi^2}\int\limits_0^\varepsilon \left( s+\frac{1}{s}- \frac{1}{2}\left(s-\frac{1}{s}\right)^2\log \left(\frac{1+s}{1-s}\right) \right)\frac{1}{s} \mbox{d} s . \end{equation}
This has a closed form, \begin{equation} \label{poly} \tilde{\chi}_1 (\varepsilon ) =\frac{2 \left(\varepsilon ^2 \left(4 \text{Li}_2(\varepsilon )-\text{Li}_2\left(\varepsilon ^2\right)\right)+\varepsilon ^4 \left(-\tanh ^{-1}(\varepsilon )\right)+\varepsilon ^3-\varepsilon +\tanh ^{-1}(\varepsilon )\right)}{\pi ^2 \varepsilon ^2}, \end{equation} where the polylogarithmic function is defined by the infinite sum \begin{equation*} \text{Li}_s (z) = \sum\limits_{k=1}^\infty \frac{z^k}{k^s}, \end{equation*} for arbitrary complex $s$ and for all complex arguments $z$ with $|z|<1$.

The further (apparently much simpler) forms above of the functions $\tilde{\chi}_2 (\varepsilon )$ and $\tilde{\chi}_4 (\varepsilon )$ were advanced in (eqs. (42), (59)), but formal proofs have not yet been developed. (Lovas and Andai did, in fact, pose the general [$\mathbb{K}= (\mathbb{R}, \mathbb{C}, \mathbb{H})$] question, but succeeded in answering it only for $\mathbb{R}$.)

The three functions $\tilde{\chi}_1 (\varepsilon ), \tilde{\chi}_2 (\varepsilon ), \tilde{\chi}_4 (\varepsilon )$ are all special cases ($d =1, 2, 4$) of the "master Lovas-Andai formula" (eq. (70) in and also in the answer in Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function) \begin{equation} \label{MasterFormula} \tilde{\chi_d}(\varepsilon)= \frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right)}{\Gamma \left(\frac{d}{2}+1\right)^2}, \end{equation} where the regularized hypergeometric function is indicated.

The normalization factor (in the notation of Lovas and Andai [no tilde now used]--their Table 2) for $d=1$ is $\chi_1(1) = \frac{2}{3} \pi^2$, and for $d=2$ is $\chi_2(1)=\frac{\pi^4}{6}$. The $\frac{2}{3} \pi^2$ value agrees (using $n = 2 d$) with that in the much-viewed mathoverflow posting Euclidean volume of the unit ball of matrices under the matrix norm . The $d=2$ $(n=4)$ value in that posting is $\frac{4}{1575 \pi^8}$, which when mulitiplied by $\frac{525 \pi^{12}}{8}$, gives us the $\frac{\pi^4}{6}$. The $d=4$ $(n=8)$ counterpart for $\mathbb{H}$ is not immediately apparent to us.

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    What does "volume in operator norm" mean? – Nik Weaver Sep 13 at 18:23
  • @NikWeaver Since the HS norm on $M_2({\bf C})$ is isometric to the usual Euclidean norm on ${\bf C}^4$ (under the standard identification of these two vector spaces), perhaps Paul means the volume as a subset of ${\bf R}^8$? – Yemon Choi Sep 13 at 23:24
  • I was perhaps inarticulately rewording the interesting Conclusion section of the Lovas-Andai paper (p. 17). I hope the since revised title and first sentence are more properly expressed than originally. – Paul B. Slater Sep 13 at 23:55
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    Paul: Nik and I know what the operator norm is. What is puzzling us is what you mean by volume – Yemon Choi Sep 14 at 23:41
  • Lovas and Andai--in the cited reference (Conclusion section)--state that they desire to compute ("a very challenging problem") the volume of the set given in the first formula of my question as a function of $\varepsilon \in [0,1]$. In their Table 2 they give the value of the function at $\varepsilon=1$ for $\mathbb{R}$ as $\frac{2}{3} \pi^2$ and for $\mathbb{C}$ as $\frac{\pi^4}{6}$. The value of $\frac{2}{3} \pi^2$ agrees with that in the (much-viewed)… – Paul B. Slater Sep 15 at 4:28

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