On Landau–Kato inequalities via semigroup orbits

Let $\omega \gt 0$. 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 $$\|e^{tA}f\|\leq e^{-\omega t}\|f\| \quad \text{and}\quad \|e^{tA}A^2f\|\leq e^{-\omega t}\|A^2f\|,\ \quad t\geq 0,$$ a dynamical inequality for $\|Af\|$ in terms of $\|f\|$ and $\|A^2f\|$ was derived by G. Herzog and P. Ch. Kunstmann [Studia Math. 223 (2014), 19–26]. 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 wondered about an analogue, again via semigroup orbits, for the Kato type inequality on Hilbert spaces. We provide such a result by using the machinery of M. Hayashi and T. Ozawa [Proc. Amer. Math. Soc. 145 (2017), 847–852], which in turn relies on Hilbertian geometry.