Paper presented at the 5th Strathmore International Mathematics Conference (SIMC 2019), 12 - 16 August 2019, Strathmore University, Nairobi, KenyaWe generalize the notion of the anti-Daugavet property (a-DP) to the anti-N-order Almasi
room, polynomial Daugavet property (a-NPDP) for Banach spaces. The characterization SBS
of the a-NPDP is through the spectral information; however, it is well-known in nonlinear
theory that there is no suitable notion of the spectra for nonlinear operators resulting into
enormous structural challenges to the known characterization techniques for the a-DP. To
bypass some of the problems, we establish that a good spectrum of a nonlinear operator is one
whose associated eigenvectors are of unit norm and study the a-NPDP for locally uniformly
convex or smooth Banach spaces (luacs); in particular, we prove that locally convex or smooth
finite dimensional Banach spaces have the a-mDP for rank-I polynomials and then extend this
result to innite dimensional luacs Banach spaces. Besides, we prove that locally uniformly
convex Banach spaces have the a-NPDP for compact polynomials if and only if their norms
are eigenvalues, and moreover, uniformly convex Banach spaces have the a-NPDP for
continuous polynomials if and only if their norms belong to the approximate spectra. As a
consequence of these results, we conclude that all continuous In-homogeneous polynomials
that satisfy the N-order polynomial Daugavet equation on a uniformly convex Banach space
such as Lr-spaces for 1 < r < 1 and Hilbert spaces have nontrivial invariant subspaces; this
result was not known.Mbarara University of Science and Technology, Ugand
