A Lower Bound for the Differences of Powers of Linear Operators

Abstract

Let T be a bounded linear operator in a Banach space, with σ(T ) = {1}. In 1983, Esterle-Berkani' s conjecture was proposed for the decay of differences (I − T ) T n as follows: Eitheror T = I. We prove this claim and discuss some of its consequences.

Microspectral analysis of quasinilpotent operators

“…In particular, it is known that the differences of consecutive powers (I−T (z))T (z) j cannot decay arbitrarily fast since either lim inf j→∞ (j + 1) (I − T (z))T (z) j ≥ 1/e or Q = 0; see [Est83,Ber83,KMSOT04,MNTY07]. The question arises whether z ∈ B Q if the fastest possible speed of decay is attained, i.e., (1.4) sup j≥1 (j + 1) (I − T (z))T (z) j < ∞.…”

On a Tauberian Condition for Bounded Linear Operators

“…for a power-bounded T satisfying σ(T ) = {1}; see also [8] (and references therein) for the determination of the optimal lower bound 1/e instead of 1/96. Hence the stronger version lim n→∞ (n+1)(I−T )T n = 0 of the tauberian condition (3) cannot generally hold for T satisfying (1) for all |λ| > 1.…”

Time regularity and functions of the Volterra operator

“…However, Esterle's result [6,Corollary 9.5] shows that the fastest decay of the differences T n+1 − T n is just O(1/n) whenever T = I and σ(T ) = {1}. Moreover, the authors of [7], [13] and [5] were able to determine the sharp constant in Esterle's result using different methods.…”