Abstract
In this paper we present the boundedness condition of the Toeplitz super-operators and we describe the quasi-Toeplitz operators over classical Bergman space with quasi-homogeneous symbols. Finally, we proved that every even super-operator acting over the Bergman super-space in the super-sphere can be written as a Toeplitz super-operator.
Highlights
Borthwick et al (1993) introduced a general theory of the non-perturbative quantization of the super-disk
In this paper we present the boundedness condition of the Toeplitz super-operators and we describe the quasiToeplitz operators over classical Bergman space with quasi-homogeneous symbols
We proved that every even super-operator acting over the Bergman super-space in the super-sphere can be written as a Toeplitz superoperator
Summary
Borthwick et al (1993) introduced a general theory of the non-perturbative quantization of the super-disk. In (Loaiza & Sanchez-Nungaray, 2010), Loaiza and Sanchez-Nungaray studied the Toeplitz operators with radial symbols acting on the Bergman space of the super-disk and they proved that every Toeplitz super-operator with radial symbol is diagonal. This result generalizes the classical case, implying that the algebra generated by all Toeplitz super-operators with radial symbols is commutative. The situation changes in the sphere because weigthed Bergman spaces are finite-dimensional In this way, Prieto-Sanabria (2009) showed that Toeplitz operators with radial symbols, acting on weigthed Bergman spaces of the sphere, are unitary equivalent to multiplication operators. In the last section we prove that every even operator over Bergman super-space on the super-sphere is a Toeplitz super-operator
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