Rudin orthogonality problem on the Bergman space |
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Authors: | Kunyu Guo |
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Affiliation: | a School of Mathematical Sciences, Fudan University, Shanghai, 200433, PR China b Center of Mathematics, Chongqing University, Chongqing 401331, PR China c Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States |
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Abstract: | In this paper, we study the Rudin orthogonality problem on the Bergman space, which is to characterize those functions bounded analytic on the unit disk whose powers form an orthogonal set in the Bergman space of the unit disk. We completely solve the problem if those functions are univalent in the unit disk or analytic in a neighborhood of the closed unit disk. As a consequence, it is shown that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity n if and only if its symbol is a constant multiple of the n-th power of a Möbius transform, which was obtained via the Hardy space theory of the bidisk in Sun et al. (2008) [10]. |
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Keywords: | Rudin?s conjecture Bergman space Multiplication operators Counting functions Orthogonal functions |
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