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How the μ-deformed Segal-Bargmann space gets two measures

Stephen Bruce Sontz — 2010

Banach Center Publications

This note explains how the two measures used to define the μ-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution...

A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

Stephen Bruce Sontz — 2013

Communications in Mathematics

We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined...

Toeplitz Quantization for Non-commutating Symbol Spaces such as $S U_{q} (2)$

Stephen Bruce Sontz — 2016

Communications in Mathematics

Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group $S U_{q} (2)$ is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new...