Spectral radii and eigenvalues of subdivision operators

Abstract

This paper discusses the spectra of matrix subdivision operators. We establish some formulas for spectral radii of subdivision operators on various invariant subspaces in ℓ p \ell _{p} . A formula for the spectral radius of a subdivision operator, in terms of the moduli of eigenvalues, is derived under a mild condition. The results are even new in the scalar case. In this case, we show that the subdivision operator has no eigenvector in ℓ p \ell _{p} if the corresponding subdivision scheme converges for some p ∈ [ 1 , ∞ ) p\in [1, \infty ) .