## Doctoral Dissertations

#### Title

On the Berezin symbol

Winter 1994

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

Let ${\cal H}$ be a functional Hilbert space of analytic functions on a complex domain $\Omega,$ with the normalized reproducing kernel function $k\sb{z},\ z\in\Omega.$ If A is a linear map of ${\cal H}$ into itself, the Berezin symbol, A, of A is defined on $\Omega$ by $\tilde{A}(z) = \langle Ak\sb{z},\ k\sb{z}\rangle.$ The purpose of this research is to study how the properties of an operator are reflected in the properties of its Berezin symbol. In summary, I have (1) studied the properties of the Berezin symbol as a complex-valued function; (2) characterized multiplication operators, induced by a multiplier of ${\cal H},$ using the multiplicative property of the map $A\mapsto\tilde{A};$ (3) expressed the boundedness and compactness properties of certain operators in terms of their Berezin symbols; and (4) characterized the set of multipliers as the set of analytic Berezin symbols of bounded operators.