 
Summary: Berezin Transform
Sheldon Axler
The Berezin transform associates smooth functions with operators on Hilbert
spaces of analytic functions. The usual setting involves an open set Cn
and
a Hilbert space H of analytic functions on . We assume that for each z ,
point evaluation at z is a continuous linear functional on H. Thus for each
z , there exists Kz H such that f(z) = f, Kz for every f H. Because
Kz reproduces the value of functions in H at z, it is called the reproducing
kernel. The normalized reproducing kernel kz is defined by kz = Kz/ Kz .
For T a bounded operator on H, the Berezin transform of T, denoted ~T, is
the complexvalued function on defined by
~T(z) = Tkz, kz .
For each bounded operator T on H, the Berezin transform ~T is a bounded
realanalytic function on . Properties of the operator T are often reflected
in properties of the Berezin transform ~T. The Berezin transform is named in
honor of F. Berezin, who introduced this concept in [4].
The Berezin transform has been useful in several contexts, ranging from the
Hardy space (see, for example, [8]) to the BargmannSegal space (see, for exam
ple, [5]), with major connections to the Bloch space and functions of bounded
