A harmonic-type maximal principle in the three chains completion problem (Q1572851)

The author stated and proved a harmonic-type maximal principle in case of commutant lifting theorem [Integral Equations Oper. Theory 28, No. 4, 373-381 (1997; Zbl 0901.47007)]. The main result of this paper is Theorem: Suppose \(\sup\{\|A_k\|: k\in\mathbb{Z}\}< 1\) and \(\|B_c\|= 1\), where \(B_c\) dnotes the central interpolant \(B(R)\) of \(A_k\) corresponding to \(R=0\). Then, \(\|B(R)\|=1\) whenever \(R\) is of norm strictly less than \(1\). Conversely, if for some \(R\) with \(|R|< 1\), we have \(\|B(R)\|= 1\), then it follows that \(\|B_c\|= 1\). Here \(A_k: H_k\to K_k\ominus M_k\) \((k\in\mathbb{Z})\), \(H_k\subset X\), \(M_k\subset K_k\subset Y\) are Hilbert spaces and \(H_{k-1}\subset H_k\), \(K_{k-1}\subset K_k\), \(M_{k-1}\subset M_k\), \(M_k\subset K_k\).