INTERPOLATION BETWEEN DOMAINS OF POWERS OF OPERATORS IN QUATERNIONIC BANACH SPACES

In contrast to the classical complex spectral theory, where thespectrum is related to the invertibility of lambda - A : D(A) subset of X-C -> X-C, in the noncommutative quaternionic S-spectral theory one uses the invertibility of the second order polynomial T-2 - 2 Re (s)T+ |s|(2 ): D(T-2) subset of 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 {te(i omega)| t > 0} ,omega is an element of [0,pi], i is an element of S is contained in the S-resolvent set, and the inverse operator (T-2-2Re(s)T+|s|(2))(-1 )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 is an element of N-0. Moreover, also a characterization of the interpolation space (X, D(T-n))(theta,p), theta is an element of (0,1), p is an element of [1,infinity], in is given in terms of integrability conditions on the pseudoS-resolvent Q(s)(-1)(T).