Paragrassmann Algebras as Quantum Spaces, Part II: Toeplitz Operators

TL;DR

This paper extends the study of paragrassmann algebras by defining and analyzing Toeplitz operators in the associated holomorphic Segal-Bargmann space, highlighting their unique properties as non-function symbols.

Contribution

It introduces and investigates Toeplitz operators in paragrassmann algebras, expanding understanding of non-commutative spaces with non-function symbols.

Findings

Toeplitz operators are defined via multiplication and projection in the Segal-Bargmann space.

These operators have symbols that are not functions.

The space itself is not a space of functions.

Abstract

This paper continues the study of paragrassmann algebras begun in Part I with the definition and analysis of Toeplitz operators in the associated holomorphic Segal-Bargmann space. These are defined in the usual way as multiplication by a symbol followed by the projection defined by the reproducing kernel. These are non-trivial examples of spaces with Toeplitz operators whose symbols are not functions and which themselves are not spaces of functions.