On Landau-Kato inequalities via semigroup orbits

TL;DR

This paper improves Landau-Kato inequalities by relaxing decay conditions on semigroup orbits and introduces a new approach to Kato-type inequalities on Hilbert spaces using Hilbertian geometry.

Contribution

It extends previous results by weakening decay assumptions and provides a new method for deriving Kato inequalities on Hilbert spaces.

Findings

Improved Landau-Kato inequalities with quadratic decay

New direct method for Landau inequality recovery

Semigroup orbit approach for Kato inequalities on Hilbert spaces

Abstract

Let $\omega>0$. Given a strongly continuous semigroup $\{e^{tA}\}$ on a Banach space and an element $f\in\mathbf{D}(A^2)$ satisfying the exponential orbital estimates $$\|e^{tA}f\|\leq e^{-\omega t}\|f\| \quad\text{and}\quad \|e^{tA}A^2f\|\leq e^{-\omega t}\|A^2f\|,\quad t\geq0,$$ a dynamical inequality for $\|Af\|$ in terms of $\|f\|$ and $\|A^2f\|$ was derived by Herzog and Kunstmann (Studia Math., 2014). Here we provide an improvement of their result by relaxing the exponential decay to quadratic, together with a simple and direct way recovering the usual Landau inequality. Herzog and Kunstmann also demanded an analogue, again via semigroup orbits, for the Kato type inequality on Hilbert spaces. We provide such a result by using Hayashi-Ozawa machinery [Proc. Amer. Math. Soc., (2017)] which in turn relies on Hilbertian geometry.