Fractional powers of higher-order vector operators on bounded and unbounded domains

Using the H-infinity-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order m >= 1, acting on the right linear quaternionic Hilbert space L-2 (Omega, C circle times H). The operators that we consider are of the type T = i(m-1) (alpha(1)(x)e(1) partial derivative(m)(x1) + alpha(2)(x)e(2)partial derivative(m)(x2) + alpha(3)(x)e(3)partial derivative(m)(x3)), x =(x(1), x(2), x(3)) is an element of (Omega) over bar, where (Omega) over bar is the closure of either a bounded domain Omega with C-1 boundary, or an unbounded domain n in R 3 with a sufficiently regular boundary, which satisfy the so-called property (R) (see Definition 1.3), e(1), e(2), e(3) is an element of H which are pairwise anticommuting imaginary units, a(1), a(2), a(3) : (Omega) over bar subset of R-3 -> R are the coefficients of T. In particular, it will be given sufficient conditions on the coefficients of T in order to generate the fractional powers of T, denoted by P-alpha (T) for alpha is an element of (0, 1), when the components of T, i.e. the operators T-l := alpha(l)partial derivative(m)(x1), do not commute among themselves. This kind of result is to be understood in the more general setting of the fractional diffusion problems. The method used to construct the fractional power of a quaternionic linear operator is a generalization of the method developed by Balakrishnan.