The Fredholm property of Toeplitz operators on the p-Fock spaces F p α on C n is studied. A general Fredholm criterion for arbitrary operators from the Toeplitz algebra Tp,α on F p α in terms of the invertibility of limit operators is derived. This paper is based on previous work, which establishes corresponding results on the unit balls B n [10].

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We consider various classes of bounded operators on the Fock space F 2 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.

Dedicated to Nikolai Vasilevski on the occasion of his 70 th birthday.

We generalize several results on Toeplitz operators over Fock spaces to the case of non-reflexive Fock spaces. Among these results are the characterization of compactness and the Fredholm property of such operators, a well-known representation of the Toeplitz algebra, a characterization of the essential centre of the Toeplitz algebra. Further, we improve several results related to correspondence theory, e.g. we improve previous results on the correspondence of algebras and we give a correspondence theoretic version of the well-known Berger-Coburn estimates.