Berezin Transform Sheldon Axler 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 complex-valued function on defined by ~T(z) = Tkz, kz . For each bounded operator T on H, the Berezin transform ~T is a bounded real-analytic 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 Bargmann-Segal space (see, for exam- ple, [5]), with major connections to the Bloch space and functions of bounded Collections: Mathematics