In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6....

This paper studies the boundary behavior of the Berezin transform on the C*-algebra generated by the analytic Toeplitz operators on the Bergman space.

We establish $L^p$-estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in ℂⁿ.

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$where $\Pi$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...

We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators $T_g$ on $H^p\left(D\right)$, where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of $T_g$ provided that $g^{-1}\left(0\right)$ is a compact subset of D.

Let $\mu$ be a positive Borel measure on the complex plane ${ℂ}^n$ and let $j=(j_1,\cdots ,j_n)$ with $j_i\in \mathbb{N}$. We study the generalized Toeplitz operators $T_{\mu }^{\left(j\right)}$ on the Fock space $F_{\alpha }^2$. We prove that $T_{\mu }^{\left(j\right)}$ is bounded (or compact) on $F_{\alpha }^2$ if and only if $\mu$ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$.

In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,\hslash )$, and gives improvements and new proofs for many of the results in the field. In the completely resonant...

We consider the solution operator $S{ℱ}_{\mu ,(p,q)}\to L^2{\left(\mu \right)}_{(p,q)}$ to the $\overline{\partial }$-operator restricted to forms with coefficients in ${ℱ}_{\mu }=\left\{ff\text{is}\text{entire}\text{and}{\int }_{{ℂ}^n}{\left|f\left(z\right)\right|}^2\mathrm{d}\mu \left(z\right)<\infty \right\}$. Here ${ℱ}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in ${ℱ}_{\mu }$, $L^2\left(\mu \right)$ is the corresponding $L^2$-space and $\mu$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\overline{\partial }$. This solution operator will have the property $Sv\perp {ℱ}_{(p,q)}\forall v\in {ℱ}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators...