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 grouprepresentation 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.

Metamorphism as a covariant transform for the SSR group

“…We start from a brief account of groups involved in the consideration. An element of the one-dimensional Heisenberg group H [10,19,25] will be denoted by (s, x, y) ∈ R 3 . The group law on H is defined as follows:…”

“…We are also using the notation ≈ f := W φ f from [26] which now can be explained as the double covariant transform for the Heisenberg and the affine groups simultaneously. The image space L φ (R 4 + ) of the metamorphism is a subspace of square-integrable functions on L 2 (R 4 + , • µ ), see (25). Remark 5.3.…”