Spectral and ergodic properties of the operator $B(r,s)$ over power series spaces ${\mathrm{\Lambda }}_{\mathrm{\infty }}(\alpha )$ of infinite type and their duals

In this paper, we investigate the spectral and ergodic properties of the linear operator $B(r,s)$ acting on power series spaces ${\mathrm{\Lambda }}_{\mathrm{\infty }}(\alpha )$ of infinite type and on their strong duals. Precisely, we provide a complete characterization of its fine spectrum and establish necessary and sufficient conditions for the operator to be power bounded and (uniformly) mean ergodic.