Power-Bounded Operators and Related Norm Estimates

Abstract

Abstract. We consider whether L = lim sup n→∞ n T n+1 − T n < ∞ implies that the operator T is power bounded. We show that this is so if L < 1/e, but it does not necessarily hold if L = 1/e. As part of our methods, we improve a result of Esterle, showing that if σ(T ) = {1} and T = I, then lim inf n→∞ n T n+1 − T n ≥ 1/e. The constant 1/e is sharp. Finally we describe a way to create many generalizations of Esterle's result, and also give many conditions on an operator which imply that its norm is equal to its spectral radius.

Subordinated discrete semigroups of operators

“…Actually, Lyubich's solution in [19] to Zemánek's question, which involves certain fractional Volterra-type operators, also seems to be a special case of Corollary 1.4 in view of the fact (see [21]) that the operator T 1 := I − V is a Kreiss operator in the spaces L p (0, 1), 1 ≤ p ≤ ∞. The following fundamental theorem on Ritt operators will be an important tool throughout the paper (for details of the theorem and further developments on Ritt operators, see [2,3,23,24,16,27,19] and their references). For θ ∈ (0, π) define the sectors Λ θ := {z ∈ C : z = 0, | Arg z| < θ} and Λ θ := {0}∪{z ∈ C : | Arg z| ≤ θ}.…”

A Lower Bound for the Differences of Powers of Linear Operators

“…That 1/e in Proposition 1 is a valid lower bound, is also proved in [6] (Kalton, MontgomerySmith, Oleszkiewicz, and Tomilov, 2002) by quite different means. Both of the existing approaches can be generalized to a larger class of results, but these respective classes are different (and we shall not discuss these generalizations here).…”

“…This proves the claim. A somewhat analogous result to the previous proposition is [6,Theorem 4.5]. We proceed to study the functions g n (z) :…”

“…Both of the existing approaches can be generalized to a larger class of results, but these respective classes are different (and we shall not discuss these generalizations here). An example is given in [6], indicating that the constant 1/e is the best possible. The construction is a modification of an example given in [7] (Lyubich, 2001).…”

On Wallen-type formulae for integrated semigroups and sine functions

“…This result has immediately been generalized: in 1967 Nakamura and Yoshida showed that (1) implies A = I in the case where A is a bounded linear operator in a Hilbert space and I is the identity operator, and one year later Hirschfeld [17] showed the same implication in the case where A is a member of a normed algebra. Later Wils [24], Chernoff [8], Nagisa and Wada [21], and Kalton et al [18] provided further related results. The approach of Wallen [23], who treated the case of a normed algebra A, and published his result even a bit before Hirschfeld, seems to be of particular simplicity and elegance.…”