Spectral projectors of bisectorial Clifford operators and applications to the generalized gradient
Abstract
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.
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BibTeX
@misc{mantovani2026spectral,
title = {{Spectral projectors of bisectorial Clifford operators and applications to the generalized gradient}},
author = {Francesco Mantovani and Peter Schlosser},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26382},
eprint = {2609.26382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26382},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral projectors of bisectorial Clifford operators and applications to the generalized gradient
AU - Francesco Mantovani
AU - Peter Schlosser
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26382
UR - https://arxiv.org/abs/2609.26382
AB - 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.
ER -