Interpolation between domains of powers of operators in quaternionic Banach spaces
Fabrizio Colombo, Peter Schlosser
Abstract
In contrast to the classical complex spectral theory, where the spectrum is related to the invertibility of $λ-A:D(A)\subseteq X_\mathbb{C}\rightarrow X_\mathbb{C}$, in the noncommutative quaternionic $S$-spectral theory one uses the invertibility of the second order polynomial $Q_s(T):=T^2-2\text{Re}(s)T+|s|^2:D(T^2)\subseteq X\rightarrow X$ 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 $\{te^{iω}\;|\;t>0\}$, $ω\in[0,π]$, $i\in\mathbb{S}$ 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 $(X,D(T^n))_{θ,p}$, $θ\in(0,1)$, $p\in[1,\infty]$, in is given in terms of integrability conditions on the pseudo $S$-resolvent $Q_s^{-1}(T)$.