Toeplitz Operators on Higher Cauchy-Riemann Spaces

We develop a theory of Toeplitz, and to some extent Hankel, operators on the kernels of powers of the boundary d-bar operator, suggested by Boutet de Monvel and Guillemin, and on their analogues, somewhat better from the point of view of complex analysis, defined using instead the covariant Cauchy-Riemann operators of Peetre and the second author. For the former, Dixmier class membership of these Hankel operators is also discussed. Our main tool are the generalized Toeplitz operators (with pseudodifferential symbols), in particular there appears naturally the problem of finding parametrices of matrices of such operators in situations when the principal symbol fails to be elliptic.

2010 Mathematics Subject Classification: Primary 47B35; Secondary 32W25, 46E20, 32W10

Keywords and Phrases: Toeplitz operator, Hankel operator, Cauchy-Riemann operators

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