Power regular operators

Abstract

We show that for a wide class of operators T T on a Banach space, including the class of decomposable operators, the sequence { ‖ T n x ‖ 1 / n } n = 1 ∞ \left \{ {{{\left \| {{T^n}x} \right \|}^{1/n}}} \right \}_{n = 1}^\infty converges for every x x in the space to the spectral radius of the restriction of T T to the subspace ∨ n = 0 ∞ { T n x } \vee _{n = 0}^\infty \{ {T^n}x\} .