Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in ℂⁿ

Let




Ω


\Omega



be a bounded pseudoconvex domain in





C

n

\mathbb {C}^n



with Lipschitz boundary and




ϕ


\phi



be a continuous function on





Ω



¯



\overline {\Omega }



. We show that the Toeplitz operator




T


ϕ




T_{\phi }



with symbol




ϕ


\phi



is compact on the weighted Bergman space if and only if




ϕ


\phi



vanishes on the boundary of




Ω


\Omega



. We also show that compactness of the Toeplitz operator




T


ϕ




p
,
q


T^{p,q}_{\phi }



on





∂



¯



\overline {\partial }



-closed




(
p
,
q
)

(p,q)



-forms for




0

≤


p

≤


n

0\leq p\leq n



and




q

≥


1

q\geq 1



is equivalent to





ϕ


=
0

\phi =0



on




Ω


\Omega



.