令H為一個separable Hilbert 空間且{e_n：n 屬於整數}為其orthonormal 基底。如果<Te_j,e_i>=c_2i-j，c_n 是一個有界Lebesgue measurable 函數 fi 在單位圓上的第 n 個Fourier series，T 就稱作slant Toepltz 算子。在[7]中已經證明T* 是一個 isometry 若且惟若|fi(z)|^2+|fi(-z)|^2 恆等於 2，若 fi 屬於 C(T) ，則T* unitarily equivalent to a shift 或to a shift and a rank one unitary，其中shift 之multiplicity 為
∞。在某些加諸&#981; 之光滑條件下， T* 相似於一個shift或一個shift、以及rank one 之unitary 的無窮multiplicity 直合的常數倍數。根據[10]，如果AS=SA且A*S=SA*那麼算子A 就是相對於一個shift 為常數的算子。因此，本文中，我們將研究相對於T為常數的算子，也就是滿足AT=TA和A*T=TA*的有界算子A。

Let H be a separable Hilbert space and {e_n : n belong to Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if <T ej , ei> =c2i&#8722;j , where c_n is the n-th Fourier series of a bounded Lebesgue measurable function on the unit circle T = {z belong to C : |z| = 1}. It has been shown [7] that T* is an isometry if and only if |fi(z)|^2 +|fi(&#8722;z)|^2 = 2 a.e. on T and if this is the case and fi belong to C(T), then either T is unitarily equivalent to a shift or to the direct sum of a shift and a rank one unitary, with infinite multiplicity
(for the shift part, that is). Moreover, with some additional assumption on the smoothness and the zeros of fi, T* is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. On the other hand, according to the terminologies in [10], an operator A that is constant with respect to a shift S if AS = SA and A S = SA . Therefore, in this article, we will study the operators that is constant with respect to T , i.e., bounded operator A satisfying AT = T A and A T = T A .

1 Introduction 4
2 The kernel of T_fi 5
3 The case fi_alpha(z) = (1 + |alpha|^2)^&#8722;1/2(alpha+z) 8

[1] D. Chen and X. Zheng, Spectral radii and eigenvalues of subdivision operators, preprint.
[2] A. Cohen and I. Daubechies, A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math. J., 1992,pp.313-335.
[3] A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Revista Mathematica Iberoamericana, 12, 1996, pp.527-591.
[4] J.B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, 36, American Mathematical Society, Providence, 1991.
[5] I. Daubechies, I. Guskov and W. Sweldens, Regularity of irregular subdivision, Constructive Approximation, 15, no. 3, 1999, pp.381-426.
[6] M.C. Ho, Adjoints of slant Toeplitz operators, Integral Equations and Operator Theory, 41, pp.179-188, 2001.
[7] M.C. Ho, Adjoints of slant Toeplitz operators II, Integral Equations and Operator Theory, 41, pp.179-188, 2001.
[8] H. Hennion, Sur un th&acute;eor`em spectral et son application noyaux Lipchiitziens, Proc. AMS, 118, pp.627-634, 1993.
[9] M. Keane, Strongly mixing g-measure, Inventiones Math., 16, pp.309-324, 1972.
[10] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, New York, 1985.
[11] G. Strang, Eigenvalues of (#2)H and convergence of the cascade algorithm, IEEE Trans. Sig. Proc., 1996.
[12] W. Sweldens and P. Schr&uml;oder, Building your own wavelets at home, Wavelets in Computer Graphics, ACMSIGGRAPH Course Notes, 1996.
[13] L. Villemoes, Wavelet analysis of refinement equations, SIAM J. Maths. Analysis, 25, no. 5, 1994, pp.1433-1460.