Hilbert Spaces of Entire Functions and Toeplitz Quantization of Euclidean Planes

• URL: http://arxiv.org/abs/2105.08400v1
• Date: Tue, 18 May 2021 09:52:48 GMT
• Title: Hilbert Spaces of Entire Functions and Toeplitz Quantization of Euclidean Planes
• Authors: Micho Durdevich and Stephen Bruce Sontz
• Abstract summary: We extend the theory of Toeplitz quantization to include diverse and interesting non-commutative realizations of the classical Euclidean plane. The Toeplitz operators are geometrically constructed as special elements from this algebra. Various illustrative examples are computed.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associated to the symbols from another quadratic non-commutative algebra, which is interpretable as polynomials over a plane to be quantized. Such a conceptual framework promotes interesting non-trivial conditions on the initial scalar product. These are analyzed in detail. Various illustrative examples are computed.

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