Dynamics of Properties of Toeplitz Operators with Radial Symbols

Abstract

Given a smooth defining symbol a=a(z), the family of Toeplitz operators $T_a = \left\{ {T_a^{\left( h \right)} } \right\}$, where h∈(0, 1), was considered in the previous chapter under the Berezin quantization procedure. For a fixed h the Toeplitz operator T _a ^(h) acts on the weighted Bergman space $\mathcal{A}_h^2 \left( \mathbb{D} \right)$, where the parameter h characterizes the weight (10.1.5) on $\mathcal{A}_h^2 \left( \mathbb{D} \right)$. In the sequel we will consider another form of presentation of the weighted Bergman spaces, see (10.1.1), the space $\mathcal{A}_\lambda ^2 \left( \mathbb{D} \right)$ which is parameterized by λ∈(−1, +∞) being connected with h∈(0, 1) by the rule $\lambda + 2 = \frac{1} {h}$, see Section 10.1.