Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group

Abstract

We introduce an extended class of cross-Toeplitz operators which act between Fock--Segal--Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc.We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time-frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time-frequency operator theory.

45 p., AMS-LateX, 3 PDF images in two figures; v2&v3: minor corrections

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Cited by in corpus (2)

• Metamorphism as a covariant transform for the SSR group
• Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle