Toeplitz operators in polyanalytic Bergman type spaces

Toeplitz and related operators on polyanalytic Fock spaces

“…The only related (but much weaker) result known to the author is for the polyanalytic Bergman space over the unit disk and due to Wolf [24]. Maybe the most unexpected result of this paper is that the compactness of Hankel operators H f,(k) on F 2 (k) does not depend on the order k. It is somewhat expected from the work of Rozenblum and Vasilevski [22] that if H f,(1) is compact, then all other Hankel operators H f,(k) are compact as well, but the other direction appears to be rather surprising in this context.…”

“…In [22] it was observed that a Toeplitz operator on a true polyanalytic Fock space F 2 (k) is unitarily equivalent to a Toeplitz operator on the analytic Fock space F 2 with a much more irregular, possibly distributional, symbol. After that observation, Rozenblum and Vasilevski offer a choice of considering "operators with nice symbols in 'bad' spaces or operators in nice spaces with 'bad' symbols".…”

“…Subsequently, several connections to time-frequency analysis, signal processing and quantum mechanics have been found. We refer the interested reader to [1,22] for now and return to related work after some introductory material.…”

“…In [22], Rozenblum and Vasilevski showed that a Toeplitz operator on a true polyanalytic Fock space F 2 (k) is unitarily equivalent to a Toeplitz operator on F 2 , but with possibly very irregular symbol. They then offer the options of either considering Toeplitz operators on 'bad' spaces with 'nice' symbols or Toeplitz operators on 'nice' spaces with 'bad' symbols.…”

Polyanalytic Toeplitz Operators: Isomorphisms, Symbolic Calculus and Approximation of Weyl Operators

“…The aim of this paper is to lift the well-established theory of Toeplitz operators to the polyanalytic setting, following initial works of Abreu, Gröchenig and Faustino [1,5,16] as well as [15,23,44]. That is, we introduce multiplication operators on Bargmann-Fock type spaces of polyanalytic functions and, thus, provide a whole new family of quantization schemes.…”