A reproducing kernel and Toeplitz operators in the quantum plane. (English) Zbl 1297.46023

Summary: 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 as Toeplitz operators. This paper extends results of the author for the finite dimensional case.


46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
81S99 General quantum mechanics and problems of quantization
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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