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- W4304889586 abstract "Abstract Using the $H^{infty }$ -functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $mgeq 1$ , acting on the right linear quaternionic Hilbert space $L^{2}(Omega,mathbb {C}otimes mathbb {H})$ . The operators that we consider are of the type begin{align*} T=i^{m-1}left(a_1(x) e_1partial_{x_1}^{m}+a_2(x) e_2partial_{x_2}^{m}+a_3(x) e_3partial_{x_3}^{m}right), quad x=(x_1,, x_2,, x_3)in overline{Omega}, end{align*} where $overline {Omega }$ is the closure of either a bounded domain $Omega$ with $C^{1}$ boundary, or an unbounded domain $Omega$ in $mathbb {R}^{3}$ with a sufficiently regular boundary, which satisfy the so-called property $(R)$ (see Definition 1.3), $e_1,, e_2,, e_3in mathbb {H}$ which are pairwise anticommuting imaginary units, $a_1,,a_2,, a_3: overline {Omega } subset mathbb {R}^{3}to mathbb {R}$ 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 $alpha in (0,1)$ , when the components of $T$ , i.e. the operators $T_l:=a_lpartial _{x_l}^{m}$ , 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." @default.
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- W4304889586 date "2022-10-13" @default.
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- W4304889586 title "Fractional powers of higher-order vector operators on bounded and unbounded domains" @default.
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- W4304889586 doi "https://doi.org/10.1017/s0013091522000396" @default.
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