The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions.

Abstract

This dissertation is devoted to the study of certain analytic problems inspired by recent results in the theory of Toeplitz operators on the Bergman spaces. We attack the problem of commuting Toeplitz operators with quasi-homogeneous symbols on the Segal-Bargmann space over $\mathbb{C}$. For a fixed monomial $z^l\overline{z}^k$ we characterize the functions $\Psi$ of polynomial growth at infinity such that the Toeplitz operators $T_\Psi$ and $T_{z^l\overline{z}^k}$ commute on the space of all holomorphic polynomials over $\mathbb{C}$. Moreover, we construct two types of commutative Banach and $C^\star$-algebras generated by Toeplitz operators acting on the Segal-Bargmann spaces over $\mathbb{C}^n$. We show that the class of symbols which generate the commutative $C^\star$-algebra of Toeplitz operators generates also a $C^\star$-algebra of operators acting on the true-$k$-Fock spaces. Furthermore, we use Toeplitz operator theory techniques to construct the heat kernel of a class of positive sub-elliptic differential operators. As an application, we obtain the heat kernel of the Grusin operator on $\mathbb{R}^{n+1}$ as well as that of the sub-Laplace operator on the $(2n+1)$-dimensional Heisenberg group ($n\in\mathbb{N}$ is arbitrary). Finally, we switch our attention to a compactness criteria for Toeplitz operators $T^{\nu}_{g}$ acting on the standard weighted Bergman spaces over bounded symmetric domains. We obtain an estimate for the Berezin tansform $\tilde{g}_{\nu_0}$ in terms of the operator norm of $T^{\nu}_{g}$ whenever $\nu$ and $\nu_0$ are suitable weights. As a consequence, we prove that for a bounded function $g$ on a bounded symmetric domain the compactness of $T^{\nu}_{g}$ is independent of the weight parameter $\nu$.