Multiplication operators on the Bergman space via analytic continuation |
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Authors: | Ronald G Douglas Shunhua Sun Dechao Zheng |
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Institution: | aDepartment of Mathematics, Texas A & M University, College Station, TX 77843, United States;bInstitute of Mathematics, Jiaxing University, Jiaxing, Zhejiang, 314001, PR China;cDepartment of Mathematics, Vanderbilt University, Nashville, TN 37240, United States |
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Abstract: | In this paper, using the group-like property of local inverses of a finite Blaschke product ?, we will show that the largest C?-algebra in the commutant of the multiplication operator M? by ? on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ?−1°? over the unit disk. If the order of the Blaschke product ? is less than or equal to eight, then every C?-algebra contained in the commutant of M? is abelian and hence the number of minimal reducing subspaces of M? equals the number of connected components of the Riemann surface of ?−1°? over the unit disk. |
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Keywords: | MSC: 47B35 30D50 46E20 |
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