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Spectral radii and eigenvalues of subdivision operators

Author: Di-Rong Chen
Journal: Proc. Amer. Math. Soc. 132 (2004), 1113-1123
MSC (2000): Primary 42C15, 47B06
Published electronically: October 9, 2003
MathSciNet review: 2045428
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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 $\ell _{p}$. 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 $\ell _{p}$ if the corresponding subdivision scheme converges for some $p\in [1, \infty )$.

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Di-Rong Chen
Affiliation: Department of Applied Mathematics, Beijing University of Aeronautics, Astronautics, Beijing 100083, China; Department of Mathematics, Hubei Institute for Nationalities, Enshi 445000, Hubei, China

Keywords: Subdivision operator, spectral radius, joint spectral radius
Received by editor(s): February 21, 2001
Received by editor(s) in revised form: December 12, 2002
Published electronically: October 9, 2003
Additional Notes: Supported in part by NSF of China under Grant 10171007 and City University of Hong Kong under Grant 7001442
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society