A Construction of Berezin–Toeplitz Operators via Schrödinger Operators and the Probabilistic Representation of Berezin–Toeplitz Semigroups Based on Planar Brownian Motion

First we discuss the construction of self-adjoint Berezin–Toeplitz operators on weighted Bergman spaces via semibounded quadratic forms. To ensure semiboundedness, regularity conditions on the real-valued functions serving as symbols of these Berezin–Toeplitz operators are imposed. Then a probabilistic expression of the sesqui-analytic integral kernel for the associated semigroups is derived. All results are the consequence of a relation of Berezin–Toeplitz operators to Schrödinger operators defined via certain quadratic forms. The probabilistic expression is derived in conjunction with the Feynman–Kac–Itô formula.