Aluthge iterations of weighted translation semigroups

The problem whether Aluthge iteration of bounded operators on a Hilbert space H is convergent was introduced in [I. Jung, E. Ko, C. Pearcy, Aluthge transforms of operators, Integral Equations Operator Theory 37 (2000) 437–448]. And the problem whether the hyponormal operators on H with dim H = ∞ has a convergent Aluthge iteration under the strong operator topology remains an open problem [I. Jung, E. Ko, C. Pearcy, The iterated Aluthge transform of an operator, Integral Equations Operator Theory 45 (2003) 375–387]. In this note we consider symbols with a fractional monotone property which generalizes hyponormality and 2-expansivity on weighted translation semigroups, and prove that if { S t } is a weighted translation semigroup whose symbol has the fractional monotone property, then its Aluthge iteration converges to a quasinormal operator under the strong operator topology.