IN THIS PAPER IS GENERALIZED THE WEIGHTED BERGMAN SPACE BW,W AND DEFINE (FORMULA) ON THE UNIT DISK U, AND WE STUDY THE COMPOSITION OPERATOR CJ ON THE BWP,W.  A COUNTEREXAMPLE FOR LEMMA 1 IN [3]. ON GENERALIZED WEIGHTED BERGMAN SPACES, COMPLEX VARIABLES, 49 (2), 109-124] IS PROVIDED AND A CORRECTED VERSION OF THE LEMMA AND CORRECTIONS ON SOME OTHER RESULTS ARE PRESENTED.

IN THIS PAPER WE PRESENT SOME CONDITIONS FOR BOUNDEDNESS AND COMPACTNESS OF THE WEIGHTED COMPOSITION OPERATORS YCJF = Y (F ◦ J) ON WEIGHTED BERGMAN SPACES AND WEIGHTED BLOCH SPACES.

WE STUDY COMPACTNESS OF VOLTERRA OPERATORS BETWEEN WEIGHTED BERGMAN SPACES BY REDUCING THIS PROBLEM TO THE COMPACTNESS OF SUITABLE MULTIPLICATION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES.

WE DETERMINE THE ADJOINTS OF MULTIPLICATION OPERATORS WITH RATIONAL SYMBOL ACTING ON THE BERGMAN AND DIRICHLET SPACES. WE USE THE RESULTS TO PRESENT THE ADJOINTS OF WEIGHTED COMPOSITION OPERATORS WITH RATIONAL SYMBOL ON MENTIONED SPACES.

FOR TWO LINEAR-FRACTIONAL SELF-MAPS OF THE UNIT DISK, WHERE AT LEAST ONE OF THEM IS A NON-AUTOMORPHISM, WE SHOW THAT THE COMMUTATOR OF COMPOSITION OPERATOR WITH THE ADJOINTS OF ANOTHER COMPOSITION OPERATOR IS NON-TRIVIALLY COMPACT ON THE WEIGHTED BERGMAN SPACES IF AND ONLY IF EITHER THESE FUNCTIONS ARE BOTH PARABOLIC OR BOTH HYPERBOLIC, WITH ASSOCIATED CONCLUSIONS ABOUT THEIR FIXED POINTS IN EACH CASE.