Level numbers of a bounded linear operator between normed linear spaces-II

We continue the study of level numbers and level vectors of a bounded linear operator between normed linear spaces. We prove that every linear operator on a finite-dimensional polyhedral Banach space possesses only finitely many level numbers. We also explore the level numbers of weighted permutation operators between $ \ell _p $ lp spaces. Furthermore, we introduce the concept of approximate level numbers and study its properties and relation with level numbers. In particular, our results indicate that the concept of level numbers can be potentially applied to the spectral theory of operators, in the setting of normed linear spaces.