The iterated Aluthge transforms of a 2-by-2 matrix converge. (Q1414718)

The Aluthge transformation takes a complex matrix \(T=U|T|\) in polar form to the matrix \(|T|^{1/2}U|T|^{1/2}\) [cf. \textit{A. Aluthge}, Integral Equations Oper. Theory 13, No. 3, 307--315 (1990; Zbl 0718.47015)]. The author proves for \(2\times 2\) complex matrices that the iterates of the Aluthge transform always converge. \textit{T. Ando} [Aluthge transforms and the convex hull of the eigenvalues of a matrix (to appear)] proved that every limit point of the sequence of iterates of the Aluthge transform on an arbitrary matrix is normal.