Paragrassmann algebras as quantum spaces, Part II: Toeplitz Operators

Sontz, Stephen Bruce

doi:10.7900/jot.2012may24.1969

Cited by 8 publications

(23 citation statements)

References 9 publications

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“…This clarifies why the examples in [22] and [23] are anti-Wick quantizations even though they arise in a non-commutative context. The longer, explicit computations given in those references are not needed as we can now see.…”

Section: Theorem 61 the Toeplitz Quantization T Is An Anti-wick Quanmentioning

confidence: 59%

“…In [22] P is the holomorphic Hilbert space, while in [23] the sub-algebra P re(θ) plays the role of P. …”

Section: Sketch Of Proofmentioning

confidence: 99%

“…The method of this section can be applied to all the examples in [22] and [23] without further ado. We shall do this later on for one of the examples from [23].…”

Section: Canonical Commutation Relationsmentioning

confidence: 99%

“…Since P ∩ P * consists either way of elements which, according to this classification, are both holomorphic and anti-holomorphic, we see the intuition behind the condition P ∩ P * = C1. But we continue with P ∩ P * being completely arbitrary.Curiously, the statement "P * is a sub-algebra of A" could be false, though it is true whenever A is a * -algebra or for the examples in [22] and [23] (even when A is not a * -algebra). In [22] P is the holomorphic Hilbert space, while in [23] the sub-algebra P re(θ) plays the role of P. Proposition 2.2 If A, P, Φ, ·, · A satisfy Conditions 1-3, then it follows that A, P * , Φ * , ·, · A satisfy Conditions 1-3.…”

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confidence: 99%

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Toeplitz Quantization for Non-commutating Symbol Spaces such as SU_q(2)

Sontz

2016

Communications in Mathematics

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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 SU 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 Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck's constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck's constant and a Hilbert space where natural, densely defined operators act.Keywords: Toeplitz quantization, non-commutating symbols, creation and annihilation operators, canonical commutation relations, anti-Wick quantization, second quantization of a quantum group MSC 2010: 47B35, 81S99 1 IntroductionThe history of Toeplitz operators covers a bit over one hundred years and includes many major works, far too numerous to mention here. For a recent reference that will give the reader some first links to that extensive literature, see Section 3.5 in [18]. Speaking for myself, the papers [6], [7] and [11] have been rather influential. But the study of Toeplitz operators with symbols coming from a non-commutative algebra seems to be limited mostly to cases where the algebra is a matrix algebra or is some other quite specific noncommutative algebra such as in two recent works, [22] and [23], of the author. The papers [22] and [23] can be considered as two rather elaborated examples of the theory presented here. That study is continued in this paper, but in a much more general setting intended to clarify the mathematical structures at play in those two examples. A new example, the quantum group SU q (2) as symbol space, will also be presented here.The paper is organized as follows. After presenting the foundations of this theory in the next section, we define and analyze in Section 3 the Toeplitz quantization. In particular, Toeplitz operators are defined as densely defined operators acting in a quantum Hilbert space. The symbols of these Toeplitz operators come from a possibly non-commutative algebra A, which in physics terminology serves as the phase space for the theory. The common domain of these Toeplitz operators is P, a pre-Hilbert space and sub-a...

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Section: Theorem 61 the Toeplitz Quantization T Is An Anti-wick Quanmentioning

confidence: 59%

“…In [22] P is the holomorphic Hilbert space, while in [23] the sub-algebra P re(θ) plays the role of P. …”

Section: Sketch Of Proofmentioning

confidence: 99%

“…The method of this section can be applied to all the examples in [22] and [23] without further ado. We shall do this later on for one of the examples from [23].…”

Section: Canonical Commutation Relationsmentioning

confidence: 99%

mentioning

confidence: 99%

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Toeplitz Quantization for Non-commutating Symbol Spaces such as SU_q(2)

Sontz

2016

Communications in Mathematics

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“…Moreover, he showed that the reproducing kernel in the Segal-Bargmann space has the standard properties. Moreover, in [9], Sontz introduced the analysis of Toeplitz operators in the associated holomorphic Segal-Bargmann space in this situation. These are defined in the usual way as multiplication by a symbol followed by the projection defined by the reproducing kernel.…”

Section: Introductionmentioning

confidence: 99%

Toeplitz operators over the paragrassmann unit disk

Pérez-Garcı́a

Sánchez-Nungaray

Pérez

2016

Bol. Soc. Mat. Mex.

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We construct a sesquilinear form on the paragrassmann super disk, this form turns out positive over the quantum Bergman space. For this quantum Bergman space we obtain the reproducing kernels and the corresponding projections. We define Toeplitz operators which are l × l matrix such that their entries are Bergman-type operators over the unit disk.Keywords Toeplitz operators · Hilbert spaces with reproducing kernels · Operators in reproducing kernel Hilbert spaces · Groups and algebras in quantum theory Mathematics Subject ClassificationPrimary 47B35; Secondary 46E22 · 47B32 · 81R05 To Sergei Grudsky on the occasion of his 60th birthday. B Armando Sánchez-Nungaray armsanchez@uv.mx Victor Pérez-García

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