Anti-N-order polynomial Daugavet property on Banach spaces

We generalize the notion of the anti-Daugavet property (a-DP) to the anti-N-order polynomial Daugavet property (a-NPDP) for Banach spaces by identifying a good spectrum of a polynomial and prove that locally uniformly alternatively convex or smooth Banach spaces have the a-mDP for rank-۱ polynomials. We then prove that locally uniformly convex Banach spaces have the a-NPDP for compact polynomials if and only if their norms are eigenvalues, and uniformly convex Banach spaces have the a-NPDP for continuous polynomials if and only if their normsbelong to the approximate spectra.

Keywords:

Banach spaces , local and uniform convexity , polynomials , N-order polynomial Daugavet equation , anti-N-order Daugavet property