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In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. In usual quantization schemes one starts with a classical phase space, then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. But we do this in the opposite order, namely the set of the eigenvalues of the previously defined annihilation operator is identified as a generalization of a classical mechanical phase space. We introduce the resolution of the identity, upper and lower symbols as well as a coherent state quantization, which in turn quantizes the Toeplitz quantization. We thereby have a curious composition of quantization schemes. We proceed by identifying a generalized Segal-Bargmann space SB of square-integrable, anti-holomorphic functions as the image of a coherent state transform. Then SB has a reproducing kernel function which allows us to define a secondary Toeplitz quantization, whose symbols are functions. Finally, this is compared with the coherent states of the Toeplitz quantization of a closely related non-commutative space known as the paragrassmann algebra., 24 pages, minor revisions, final version

This article revolves around the properties on the Lp scale of spaces of the integral kernel operator K whose kernel function is the reproducing kernel of the Segal-Bargmann space. We find sufficient conditions on p and q for K to be a Hille-Tamarkin (and hence compact) operator from Lp to Lq with respect to the standard Gaussian measure as well as with respect to a weighted measure on the codomain space. We also find sufficient conditions for K to be unbounded with respect to the standard Gaussian measure. Finally we give sufficent conditions for a Toeplitz operator to be Hille-Tamarkin on the Lp scale of spaces with respect to both the standard Gaussian measure and a weighted measure on the codomain space.

This article is based on recent results of the author on the properties of the reproducing kernel of the Segal-Bargmann space. Those results are used here to demonstrate a family of energy-entropy inequalities in the Segal-Bargmann space. In some cases this is a log-Sobolev inequality while in other cases this is actually a reverse log-Sobolev inequality, which means that the energy term is bounded above by the entropy term, plus a norm term. This implies that in the Segal-Bargmann space the entropy is finite if and only if the energy is finite. Applications of this result to the Segal-Bargmann transform are given as well as a discussion of its possible relation with reverse hypercontractivity. It should be noted that all of the results of this article are proved without using hypercontractivity estimates.

The Segal-Bargmann transform A is a unitary isomorphism of the quantum mechanical configuration space L2(Rn,ν) onto the Segal-Bargmann, or phase, space HL2(Cn,μ) of entire functions square integrable with respect to μ. Here μ and ν are Gaussian measures, and for z∈Cn and measurable f:Rn→CAf(z) :=∫Rn dν(t)f(t)exp(−z2/2+√z⋅t) if the integral converges. Af is defined for all f∈Lp(Rn,ν) if 1 1+q/2 (which implies p>q) and in this case ‖A‖p→q⩽((1+q/2−p′q/2)(1−q/2))−n/2q where p′ is the dual of p. Moreover, a calculation with test functions shows that ‖A‖p→q=∞ if q>2 or p 1+q/2, Cp,q is given explicitly as a function of p and q and S...

We prove reverse hypercontractivity inequalities as well as reverse log-Sobolev inequalities in the context of a space of holomorphic functions, which is called the μ-deformed Segal-Bargmann space and arises in the works of Wigner, Rosenblum, and Marron. To achieve this we define μ-deformations of energy and entropy. Our principle results generalize earlier works of Carlen and Sontz. We also show that the semigroup of this theory is Lp bounded, and we conjecture that it is Lp contractive and, even more strongly, that it is hypercontractive.

We consider a μ-deformation of the Segal-Bargmann transform, which is a unitary map from a μ-deformed quantum configuration space onto a μ-deformed quantum phase space (the μ-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse log-Sobolev inequalities in the μ-deformed Segal-Bargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.