Some results about Beurling algebras with applications to operator theory

We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying ${\displaystyle \parallel T^n\parallel =O\left(n^{\beta }\right)}$ as n → ∞ for some β ≥ 0, then ${\displaystyle {\sum }_{n=1}^{\infty }\parallel {(1-T)}^{n}x\frac{\parallel }{\parallel }{(1-T)}^{n-1}x\parallel }$ diverges for every x ∈ X such that ${\displaystyle {(1-T)}^{[\beta ]+1}x\ne 0}$.