On the Crank-Nicolson scheme once again (Q657033)

Let \((\tau_j)_{j\in\mathbb{N}}\) be a sequence of strictly positive real numbers, and let \(A\) be the generator of a bounded analytic semigroup in a Banach space \(X\). Put \(A_n=\prod_{j=1}^n(I+\frac{1}{2} \tau_jA) (I-\frac{1}{2} \tau_jA)^{-1}\), and let \(x\in X\). Define the sequence \((x_n)_{n\in\mathbb{N}}\subset X\) by the Crank-Nicolson scheme: \(x _n = A_n x\). In this paper, it is proved that the Crank-Nicolson scheme is stable in the sense that \(\sup_{n\in\mathbb{N}}\| A_nx\|<\infty\). A serious flaw affects the main stability result. The author gives the correct formulation in an Erratum [\textit{J. A. van Casteren}, ``Erratum to: On the Crank-Nicolson scheme once again'', J. Evol. Equ. 11, No. 2, 477--483 (2011; Zbl 1239.47034)].