Rates of Decay for $(α, β)$-Ritt-Kreiss Operators

Abstract: This paper investigates the growth of the sequences $(|T^{n(I-T)^k|)_{n\ge} 1}$ for bounded linear operators on Banach spaces under various resolvent conditions. Focusing on $α$-Ritt operators that are also strongly Kreiss bounded, we show that the growth estimates established by Nevanlinna for Kreiss bounded operators can be significantly refined within the Hilbert space setting, nearly attaining the optimal rates obtained by Seifert for the more restrictive power-bounded case. We further introduce the class of $(α, β)$-RK operators as a generalization of both Ritt and Kreiss-type conditions. For these operators, we derive comprehensive growth estimates which, for certain ranges of $α$ and $β$, yield improvements over existing bounds in the literature. Particular attention is given to $β$-Kreiss operators, for which we provide a characterization via Cesàro-type means. We show that, in contrast to the well-known case $β= 1$, the power growth estimate $|T^n| = O(n<sup>β)$ is sharp whenever $β&gt; 1$. The optimality of our estimates is discussed in several cases, relying on constructions and techniques developed by Nevanlinna, Spijker, and Borovykh. We conclude by providing a characterization of Ritt operators that appears to be absent from the literature.

• The paper establishes sharp asymptotic decay bounds for operator sequences using resolvent estimates across Banach and Hilbert spaces.
• It introduces a novel (α,β)-Ritt-Kreiss framework that generalizes classical Ritt and Kreiss operators through advanced functional calculus and moment inequalities.
• The results yield refined growth estimates and equivalences that have significant implications for stability in numerical methods and ergodic operator theory.

Rates of Decay for $(\alpha, \beta)$-Ritt-Kreiss Operators

Introduction

The paper "Rates of Decay for $(\alpha, \beta)$-Ritt-Kreiss Operators" (2606.14323) develops sharp quantitative asymptotic bounds for the sequences $\|T^n(I-T)^k\|$ for various classes of bounded operators $T$ acting on Banach and Hilbert spaces, under resolvent-type assumptions. The framework encompasses Ritt, Kreiss, and newly defined $(\alpha, \beta)$-Ritt-Kreiss operators, generalizing classical decay theorems and extending current bounds on norm growth. The analysis leverages resolvent estimates, functional calculus, interpolation inequalities, and Cesàro-type means.

Classical and Generalized Operator Classes

The Katznelson-Tzafriri theorem provides qualitative convergence guarantees for power-bounded operators $T$ when $\sigma(T)\cap\mathbb{T}\subset\{1\}$; quantitative decay bounds require stronger spectral or resolvent constraints. An $\alpha$-Ritt operator is defined by the resolvent estimate $\|R(\lambda, T)\|\le C|\lambda-1|^{-\alpha}$ for $1<|\lambda|<2$. Kreiss boundedness, $(\alpha, \beta)$0, is strictly weaker than power-boundedness in infinite dimensions, and strong Kreiss boundedness ($(\alpha, \beta)$1) yields more refined norm growth bounds tied to Banach space geometry.

The paper introduces $(\alpha, \beta)$2-RK operators, defined via $(\alpha, \beta)$3 for $(\alpha, \beta)$4, subsuming both Ritt ($(\alpha, \beta)$5) and Kreiss ($(\alpha, \beta)$6) types. For these operators, norm decay rates depend on $(\alpha, \beta)$7, $(\alpha, \beta)$8, and the interplay with operator moments.

Decay Rates and Main Results

For power-bounded $(\alpha, \beta)$9-Ritt operators, Seifert established $\|T^n(I-T)^k\|$0, recovering Ritt's optimal $\|T^n(I-T)^k\|$1 for $\|T^n(I-T)^k\|$2. The paper significantly refines Nevanlinna's earlier estimates for Kreiss bounded $\|T^n(I-T)^k\|$3-Ritt operators by demonstrating that, under strong Kreiss boundedness and in Hilbert spaces, an analogous rate holds:

$\|T^n(I-T)^k\|$4

The new $\|T^n(I-T)^k\|$5-RK framework allows the authors to prove—for certain $\|T^n(I-T)^k\|$6—that

$\|T^n(I-T)^k\|$7

for $\|T^n(I-T)^k\|$8, generalizing and improving prior results, e.g., those of Mahillo and Rueda. For $\|T^n(I-T)^k\|$9-Kreiss operators with $T$0, they rigorously characterize the growth of Cesàro-type means, showing $T$1 is sharp: explicit constructions achieve $T$2. This assertion sharply contrasts with the classical $T$3 case ($T$4 in Hilbert spaces).

The paper establishes equivalence between uniform Cesàro boundedness and $T$5-Kreiss boundedness for positive operators on Banach lattices, extending classical results beyond $T$6.

Proof Techniques and Structural Innovations

Rigorous technical developments include:

• Resolvent contour integration: Dunford-Riesz calculus is used to represent $T$7 and analyze the decay using resolvent bounds, leveraging function-theoretic estimates.
• Moment inequalities and functional calculus: For fractional powers $T$8, interpolation yields sharper bounds, especially in Hilbert and UMD spaces.
• Detailed asymptotics: Subharmonicity and maximum principle arguments provide uniform bounds on resolvent curves.
• Operator-theoretic constructions: Examples from Nevanlinna, Borovykh-Spijker, and Bonilla-Müller concretely demonstrate sharpness of bounds and necessity of log-factors.

Contradictory and Strong Claims

The paper explicitly demonstrates the sharpness of the $T$9 growth for $(\alpha, \beta)$0-Kreiss operators ($(\alpha, \beta)$1), contradicting the naive expectation that $(\alpha, \beta)$2: constructed operators (even positive ones) achieve these rates exactly. Similarly, for $(\alpha, \beta)$3-RK operators (with $(\alpha, \beta)$4), the authors provide bounds that improve over previous literature and show that certain log-growth factors are necessary.

Practical and Theoretical Implications

The results directly inform discrete functional analysis, operator ergodic theory, and numerical analysis (especially stability of iterative and splitting schemes for differential equations). In mathematical analysis, these results clarify the precise role resolvent conditions play in norm decay and their sensitivity to geometric, spectral, and functional properties. For applications in numerical methods, the growth rates serve as thresholds for algorithmic stability and error propagation in iterative solvers.

Potential future impact includes:

• Extending these decay bounds to more general operator semigroups and noncommutative $(\alpha, \beta)$5 spaces.
• Deeper characterization of the boundary cases of $(\alpha, \beta)$6-RK operators, e.g., open questions regarding sharpness for $(\alpha, \beta)$7.
• Further refinement in AI-related areas: abstract operator stability often governs convergence of iterative algorithms in machine learning, optimization, and applied spectral methods.

Conclusion

This paper rigorously advances decay estimates for operator sequences in Banach and Hilbert spaces, systematically generalizing Ritt and Kreiss frameworks to $(\alpha, \beta)$8-RK operators. It refines classical bounds, demonstrates sharpness with constructive examples, and establishes equivalence of Cesàro and Kreiss boundedness in extended settings. The results carry significant practical and theoretical consequences for operator theory, ergodic analysis, and stability of numerical methods (2606.14323).