Interpolation between domains of powers of operators in quaternionic Banach spaces

In contrast to the classical complex spectral theory, where the spectrum is related to the invertibility of λ-A:D(A)⊆ XC→ XC, in the noncommutative quaternionic S-spectral theory one uses the invertibility of the second order polynomial Qₛ(T):=T²-2Re(s)T+|s|²:D(T²)⊆ X→ 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 ω\;|\;t>0\, ω∈[0,π], i is contained in the S-resolvent set, and the inverse operator Qₛ⁻¹(T) admits certain decay properties on this ray. Utilizing the K-interpolation method, we then demonstrate that the domain D(Tᵏ) of the k-th power of T is an intermediate space between D(Tⁿ) and D(Tᵐ), whenever n<k<m₀. Moreover, also a characterization of the interpolation space (X,D(Tⁿ))θ,p, θ∈(0,1), p∈[1,∞], in is given in terms of integrability conditions on the pseudo S-resolvent Qₛ⁻¹(T).