A construction of Berezin-Toeplitz operators via Schrodinger 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 sernigroups is derived. All results are the consequence of a relation of Berezin-Toeplitz operators to Schrodinger operators defined via certain quadratic forms. The probabilistic expression is derived in conjunction with the Feynman-Kac-It (o) over cap formula.