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

Abstract

Using the H ∞ -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 (Ω, C ⊗ H). The operators that we consider are of the typewhere Ω is the closure of either a bounded domain Ω with C 1 boundary, or an unbounded domain Ω in R 3 with a sufficiently regular boundary which satisfy the so called property (R) (see Definition 1.3), {e1, e2, e3} is an orthonormal basis for the imaginary units of H, a1, a2, a3 : Ω ⊂ 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 Pα(T ) for α ∈ (0, 1), when the components of T , i.e. the operators T l := a l ∂ m x l , do not commute among themselves.

Axially harmonic functions and the harmonic functional calculus on the S-spectrum

“…Nowadays there are several research directions in the area of the spectral theory on the Sspectrum, and without claiming completeness we mention: the characteristic operator function, see [1], slice hyperholomorphic Schur analysis, see [5], and several applications to fractional powers of vector operators that describe fractional Fourier's laws for nonhomogeneous materials, see for example [6,17,18]. These results on the fractional powers are based on the H ∞ -functional calculus (see the seminal papers [4], [14]).…”