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Proceedings of the American Mathematical Society

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Spectra of subdivision operators

Author: Ding-Xuan Zhou
Journal: Proc. Amer. Math. Soc. 129 (2001), 191-202
MSC (1991): Primary 42C15, 47B35
Published electronically: June 21, 2000
MathSciNet review: 1784023
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Let $a:=\{ a(k)\}_{k\in \mathbb{Z}}$ be a sequence of complex numbers and $a(k)=0$ except for finitely many $k$. The subdivision operator $S_{a}$ associated with $a$ is the bi-infinite matrix $S_{a}:= \left ( a(j-2k)\right )_{j, k\in \mathbb{Z}}$. This operator plays an important role in wavelet analysis and subdivision algorithms. As the adjoint it is closely related to the well-known transfer operators (also called Ruelle operator).

In this paper we show that for any $1\le p\le \infty $, the spectrum of $S_{a}$ in $\ell _{p}(\mathbb{Z})$ is always a closed disc centered at the origin. Moreover, except for finitely many points, all the points in the open disc of the spectrum lie in the residual spectrum.

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Additional Information

Ding-Xuan Zhou
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Keywords: Subdivision operator, spectrum, residual spectrum, wavelet analysis, joint spectral radius
Received by editor(s): June 24, 1998
Received by editor(s) in revised form: March 31, 1999
Published electronically: June 21, 2000
Additional Notes: This research was supported in part by Research Grants Council of Hong Kong
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society