We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.

中文翻译：

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多解析Toeplitz算子：同构，符号演算和Weyl算子的逼近

我们讨论基于多元分析函数的Toeplitz量化的扩展。我们推导了多解析函数的加权Sobolev-Fock空间之间的多解析Toeplitz算子的同构定理，这些空间是多解析Bargmann变换下的调制空间图像。这归纳了分析设置中众所周知的结果。最后，我们导出了渐近符号演算，并根据多解析Toeplitz算子给出了复杂Weyl算子的渐近展开。