Characterization of decay rates for discrete operator semigroups

2024/12/27 by Masashi Wakaiki, Wakaiki, Masashi
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Advanced Mathematical Modeling in Engineering #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2412.19534

Abstract

Let T be a power-bounded linear operator on a Hilbert space X, and let S be a bounded linear operator from another Hilbert space Y to X. We investigate the non-exponential rate of decay of ‖TⁿS‖ as n → ∞. First, when X = Y and S commutes with T, we characterize the decay rate of ‖TⁿS‖ in terms of the growth rate of ‖(λI - T)^-kS‖ as |λ| \downarrow 1 for some k ∈ ℕ. Next, we provide another characterization by means of an integral estimate of ‖(λI - T)^-kS‖. The second characterization is then applied to asymptotic estimates for perturbed discrete operator semigroups. Finally, we present some results on the relation between the decay rate of ‖TⁿS‖ and the boundedness of the sum ∑_n=1^∞ f(n)‖TⁿSy‖^p for all y ∈ Y in the Banach space setting, where f \colon ℕ →(0,∞) and p ≥ 1.

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