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

DOI10.1007/S43034-022-00249-7arXiv2108.13710MaRDI QIDQ6376446

Publication date: 31 August 2021

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.

Mathematics Subject Classification ID

Convolution as an integral transform (44A35) Toeplitz operators, Hankel operators, Wiener-Hopf operators (47B35) Coherent states (81R30) Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics (81S30) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22) (L^p)-spaces and other function spaces on groups, semigroups, etc. (43A15) Pseudodifferential operators (47G30) Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) (47B32) Bergman spaces and Fock spaces (30H20)

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