On the extended matrix disc conjecture (Q1201563)

Let \(\Delta\) denote the set of all complex \(2\times 2\)-matrices \(Z\) such that \(I-ZZ^*\) is positive definite. Define \[ H_ n:=\{(Z_ 1B,\dots,Z_ nB):Z_ 1,\dots,Z_ n\in\Delta,\quad B\in SL(2,\mathbb{C})\}, \] \[ D_ n:=\{(AZ_ 1B,\dots,AZ_ nB):Z_ 1,\dots,Z_ n\in\Delta,\quad A,B\in SL(2,\mathbb{C})\}. \] The author proves that \[ H_ 2=\{(W_ 1,W_ 2)\in\mathbb{C}^ 2[2\times 2]:W_ 1W_ 2^*\in\Delta,\quad|\text{det} W_ 1|<1,\;|\text{det} W_ 2|<1\}. \] Using this result he shows that \(H_ 2\) and \(D_ 2\) are holomorphically convex. Notice that the fact that \(D_ n\) is holomorphically convex for arbitrary \(n\) was proved by \textit{A. G. Sergeev} and \textit{P. Heinzler} [Izv. Akad. Nauk SSSR, Ser. Mat. 55, 647-657 (1991; Zbl 0754.32006)].