arXiv2026
We consider right-linear operators $T$ on a right Hilbert module $V$ over the Clifford algebra R_n, whose S-spectrum lies in an acute double sector. For these bisectorial operators, we introduce spectral projectors P_\pm associated with the two cones of the double sector. They decompose the Hilbert module into two submodules V=V_++V_-, and the bisectorial operator T into two sectorial operators T|_\pm. A crucial but non-trivial, cornerstone in this theory is the boundedness of the projectors P_\pm, which is, in turn connected to a bounded H^\infty-functional calculus of the operator T. We provide two practical criteria: either the squared operator admits a bounded H^\infty-functional calculus, or the operator is m-accretive. Finally, we apply these results to the gradient operator \nabla_a with nonconstant coefficients. For the particular gradient with constant coefficients, we are even able to derive explicit representations of the submodules V_\pm and the projectors P_\pm in Fourier space. Moreover, we identify the sign of the gradient operator with the Clifford-Hilbert transform. This sign plays a central role in the fractional powers of vector operators, which are used, for instance, in the non-local Fourier law of heat propagation.