We will prove local and global Besov-type characterisations for the Bloch space and the little Bloch space. As a special case we obtain that the Bloch space consists of those analytic functions on the unit disc whose restrictions to pseudo-hyperbolic discs (of fixed pseudo-hyperbolic radius) uniformly belong to the Besov space. We also generalise the results to Bloch functions and little Bloch functions on the unit ball in C m . Finally we discuss the related spaces BMOA and VMOA.
Abstract. In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline.
We will discuss invertibility of Toeplitz products T f T % g g ; for analytic f and g; on the Bergman space and the Hardy space. We will furthermore describe when these Toeplitz products are Fredholm. # 2002 Elsevier Science (USA)
The Bourgain algebra of H°°(Ώ) relative to L°°(B) is shown to be H°°(B) + C(l) + V, where V is an ideal of functions in L°°(ID)) which vanish in an appropriate sense near the boundary of ID).
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.
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