On the multiplicities of the powers of a Banach space operator

Abstract:

The multiplicities of the powers of a bounded linear operator $T$, acting on a complex separable infinite-dimensional Banach space $\mathfrak {X}$, satisfy the inequalities \[ ( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1.\] Nothing else can be said, in general, because simple examples show that for each sequence $\{ {\mu _n}\} _{n = 1}^\infty$, satisfying the inequalities $( * * )$, there exists $T$ acting on $\mathfrak {X}$ such that $\mu ({T^n}) = {\mu _n}$ for all $n \geqslant 1$.