On Landau-Kato inequalities via semigroup orbits

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 $$\Vert e^{tA}f\Vert \le e^{-\omega t}\Vert f\Vert \text{and}\Vert e^{tA}{A}^{2}f\Vert \le e^{-\omega t}\Vert A^{2}f\Vert ,t\ge 0,$$ a dynamical inequality for $\Vert Af\Vert$ in terms of $\Vert f\Vert$ and $\Vert A^{2}f\Vert$ 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.

added a section on Kato inequality in the Hilbertian case, hence the title changed