Reducing problem of multiplication operators on Bergman space

THE L~2(D) refers to Bergman space on D, where D is the unit disk on the complex plane. Using the super-isometric dilation technique, we obtain the following results. Proposition 1. The multiplication operator Mon Bergman space L~2 (D) is unitarily equivalent to the compression of the direct sum of 2N-1 copies of Bergman shift, where φ is a Blaschke product of order N (<∞).