Abstract
Let Bn denote the open unit ball in
Cn. We write V to denote Lebesgue
volume measure on Bn normalized so that V(Bn)=1.
Fix
−1<γ<∞ and let Vγ denote
the measure given by
dVγ(z)=cγ
(1−[mid ]z[mid ]2)γdV(z),
for z∈Bn, where
cγ=Γ(n+γ+1)/
(n!Γ(γ+1)); then
Vγ(Bn)=1. The weighted
Bergman space A2,γ(Bn)
is the space of all analytic functions in
L2(Bn, dVγ).
This is a closed linear subspace of
L2(Bn, dVγ).
Let Pγ denote the orthogonal projection of
L2(Bn, dVγ)
onto
A2,γ(Bn). For a
function
f∈L∞(Bn)
the Toeplitz operator Tf is defined on
A2,γ(Bn)
by Tfh=Pγ(fh),
for h∈A2,γ(Bn).
It is clear that Tf is bounded on
A2,γ(Bn) with
∥Tf∥[les ]∥f∥∞.
In this paper we will consider the question for which
f∈L∞(Bn)
the operator Tf is compact on
A2,γ(Bn).
Although a complete answer has been
given by the author and D. Zheng (see the next section), the condition
for
compactness is somewhat unnatural. In this article we will give a more
natural
description for compactness of Toeplitz operators with sufficiently nice
symbols. We will
describe compactness in terms of behaviour of the so-called Berezin
transform of
the symbol, which has been useful in characterizing compactness of Toeplitz
operators
with positive symbols (see [5, 9]).
Before we can define this Berezin transform we need to introduce more notation.
Publisher
Cambridge University Press (CUP)
Cited by
26 articles.
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