Interpolation between domains of powers of operators in quaternionic Banach spaces

Abstract

In contrast to the classical complex spectral theory, where the spectrum is related to the invertibility of $\lambda -A:D(A)\subseteq X_{\mathbb{C}}\to X_{\mathbb{C}}$, in the noncommutative quaternionic $S$-spectral theory one uses the invertibility of the second order polynomial $Q_s(T):=T^2-2\text{Re}(s)T+|s{|}^2:D(T^2)\subseteq X\to X$ to define the $S$-spectrum, where $X$ is a quaternionic Banach space. In this paper we will consider quaternionic operators $T$, for which at least one ray $\{t{e}^{i\omega }|t>0\}$, $\omega \in [0,\pi ]$, $i\in \mathbb{S}$ is contained in the $S$-resolvent set, and the inverse operator $Q_s^{-1}(T)$ admits certain decay properties on this ray. Utilizing the $K$-interpolation method, we then demonstrate that the domain $D(T^k)$ of the $k$-th power of $T$ is an intermediate space between $D(T^n)$ and $D(T^m)$, whenever $n<k<m\in {\mathbb{N}}_0$. Moreover, also a characterization of the interpolation space $(X,D(T^n){)}_{\theta ,p}$, $\theta \in (0,1)$, $p\in [1,\infty ]$, in is given in terms of integrability conditions on the pseudo $S$-resolvent $Q_s^{-1}(T)$.