Multiplication Operator with BMO Symbols and Berezin Transform

We discuss multiplication operator with a special symbol on the weighted Bergman space of the unit ball. We give the necessary and sufficient conditions for the compactness of multiplication operator on the weighted Bergman space of the unit ball.

Given  ∈ respectively.For  ∈  1 (  , V  ), we define the Berezin transform of  to be the function f; that is, ( For  ∈   , let   be the automorphism of   such that   (0) =  and   = (  ) −1 .Thus, we have the change-ofvariable formula for every ℎ ∈  1 (  , V  ).
Multiplication operators are one of the most widely studied classes of concrete operators.The study of their behavior on the Hardy and Bergman spaces has generated an extensive list of results in the operator theory and in the theory of function spaces [1][2][3][4][5][6].One of the useful approaches is the use of the Berezin transform [7][8][9][10][11].This method is motivated by its connections with quantum physics and noncommutative geometry.
In general, Berezin transform f plays important role in giving necessary and sufficient conditions for the boundedness and compactness of the Toeplitz operator [12,13].However Berezin transform || 2 or the mean oscillation MO() is used to obtain the necessary and sufficient conditions for the boundedness and compactness of the Hankel operator or multiplication operator [14,15].This work is partially motivated by using Berezin transform f to obtain necessary and sufficient conditions for the compactness of multiplication operator on the weighted Bergman space of the unit ball.
Throughout the paper, we will use the letter  to denote a generic positive constant that can change its value at each occurrence.

Main Results
In this section, we give the necessary and sufficient conditions for the compactness of multiplication operator on the weighted Bergman space of the unit ball.We furthermore obtain the necessary and sufficient conditions for the compactness of Toeplitz operator and Hankel operator. Since it is clear that (  ) * =   .It suffices to prove that the operator (  ) * is compact by showing that (  ) * can be approximated by compact operators in the operator norm.