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Abstract
In this paper, we investigate compactness of the commutator on the Hardy space
or the weighted Bergman space
(
), when
and
are automorphisms of the unit ball
. We obtain that
is compact if and only if both
and
are unitary and they commute. This generalizes the corresponding result in one variable. Moreover, our technique is different and simple. In addition, we also discuss the commutator
on the Dirichlet space
, where
and
are linear fractional self-maps or both automorphisms of
.
AMS Subject Classifications:
Acknowledgements
During the author’s visit at The College at Brockport, State University of New York, they provided a good environment for working on this paper. Shanghai Municipal Education Commission provided the financial support during her visiting. She would like to express her gratitude to them. The author also thanks the referees for giving lots of constructive suggestions and for pointing out the Ref. [Citation19].
Notes
No potential conflict of interest was reported by the author.