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We consider various classes of bounded operators on the Fock space F² of Gaussian square integrable entire functions over the complex plane. These include Toeplitz (type) operators, weighted composition operators, singular integral operators, Volterra-type operators and Hausdorff operators and range from classical objects in harmonic analysis to more recently introduced classes. As a leading problem and closely linked to well-known compactness characterizations we pursue the question of when these operators are contained in the Toeplitz algebra. This paper combines a (certainly in-complete) survey of the classical and more recent literature including new ideas for proofs from the perspective of quantum harmonic analysis (QHA). Moreover, we have added a number of new theorems and links between known results.
Bauer et al. (Fri,) studied this question.