Segal-Bargmann Space Research Articles

Cross-Toeplitz operators on the Fock–Segal–Bargmann spaces and two-sided convolutions on the Heisenberg group

We introduce an extended class of cross-Toeplitz operators which act between Fock–Segal–Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc. We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time–frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in a detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time–frequency operator theory.

Self-adjointness of Toeplitz operators on the Segal-Bargmann space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.

Segal–Bargmann Transforms Associated to a Family of Coupled Supersymmetries

The Segal–Bargmann transform is a Lie algebra and Hilbert space isomorphism between real and complex representations of the oscillator algebra. The Segal–Bargmann transform is useful in time-frequency analysis as it is closely related to the short-time Fourier transform. The Segal–Bargmann space provides a useful example of a reproducing kernel Hilbert space. Coupled supersymmetries (coupled SUSYs) are generalizations of the quantum harmonic oscillator that have a built-in supersymmetric nature and enjoy similar properties to the quantum harmonic oscillator. In this paper, we will develop Segal–Bargmann transforms for a specific class of coupled SUSYs which includes the quantum harmonic oscillator as a special case. We will show that the associated Segal–Bargmann spaces are distinct from the usual Segal–Bargmann space: their associated weight functions are no longer Gaussian and are spanned by stricter subsets of the holomorphic polynomials. The coupled SUSY Segal–Bargmann spaces provide new examples of reproducing kernel Hilbert spaces.

Cyclic Composition operators on Segal-Bargmann space

Abstract We study the cyclic, supercyclic and hypercyclic properties of a composition operator C ϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ(z) = Az + b, A is a bounded linear operator on ℰ, b ∈ ℰ with ||A|| ⩽ 1 and A*b belongs to the range of (I – A * A)½. Specifically, under some conditions on the symbol ϕ we show that if C ϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if C ϕ * is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ.

The ∂-complex on weighted Bergman spaces on Hermitian manifolds

In this paper, we investigate the ∂-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in Cn, studied earlier by the first-named author, in which the adjoint of the differentiation is the multiplication by z. The results are applied to two important examples in the unit ball, namely, the complex hyperbolic metric and a conformally Kähler metric which are related to Bergman spaces with so-called “exponential” and “standard” weights, respectively. In particular, we obtain new estimates for the solutions of the ∂-equation on these weighted Bergman spaces.

Coherent states for the Manin plane via Toeplitz quantization

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 and then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. However, 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, and 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.

Compact vectorial Toeplitz operators on the Segal-Bargmann space

We provide a sufficient condition for the compactness of a Toeplitz operator acting on the Segal-Bargmann space of vector-valued functions written in terms of an associated operator-valued kernel.

Construction of some classes of commutative Banach and C⋆-algebras of Toeplitz operators

In the present paper, we construct commutative algebras generated by Toeplitz operators on the Segal–Bargmann space Hs2(Cn) and on the true-k-Fock spaces. Analogous to the commutative Banach algebras constructed by N. Vasilevski for the case of the unit ball, we obtain a commutative algebra on Hs2(Cn) formed only by Toeplitz operators and a composition formula is obtained. Employing a natural extension for the notion of Toeplitz operators, we introduce “true-k-Toeplitz operators” acting on the true-k-Fock spaces. We provide a commutative C⋆-algebra generated by such operators, whose symbol depends on the real and imaginary parts of the complex variable in a certain sense.

The $\partial $-complex on the Segal–Bargmann space

The $\partial $-complex on the Segal–Bargmann space

Holomorphic Hermite Functions in Segal–Bargmann Spaces

We study systems of holomorphic Hermite functions in the Segal–Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose annihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.

Composition operators between Segal-Bargmann spaces

Composition operators between Segal-Bargmann spaces

Vanishing Bergman Kernels on the Disk

We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal–Bargmann spaces of the complex plane are briefly discussed.

Holomorphic Hermite functions and ellipses

ABSTRACTIn 1990 van Eijndhoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand–Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present paper shows that their systems of holomorphic Hermite functions are determined by some cases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal–Bargmann spaces determined by the Bargmann-type transforms introduced by Sjöstrand.

Composition operators on Hilbert spaces of entire functions with analytic symbols

Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Carswell, MacCluer and Schuster is generalized to the case of Segal–Bargmann spaces of infinite order. Some related questions are also discussed.

Inglés

Using standard techniques from geometric quantization, we re- derive the product of functions on R2 which was first introduced by von Neumann and later reintroduced by Groenewold and which is the integral version of the Moyal product. More specifically, by pairing the diagonal real polarization on the pair groupoid R2 × R2 with its standard holomorphic polarization, we obtain the well-known Segal-Bargmann transform in a ro tated and scaled (and half-conjugated) form. Together with a convolution of functions in the Segal-Bargmann space, which is a natural deformation of the usual convolution of functions on the pair groupoid, this defines the Groenewold-von Neumann product on L2(R2). Keywords : Geometric quantization, star product, Segal-Bargmann trans form, Fock spaces. To cite this article: J.B. Moreno, Groenewold-von Neumann product via Segal-Bargmann transform, Rev. Integr. Temas Mat. 33 (2015), No. 2, 135–144.

Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics

Continuing our earlier investigation of the Hermite case [S. T. Ali and M. Engliš, J. Math. Phys. 55, 042102 (2014)], we study an unorthodox variant of the Berezin-Toeplitz quantization scheme associated with Laguerre polynomials. In particular, we describe a “Laguerre analogue” of the classical Fock (Segal-Bargmann) space and the relevant semi-classical asymptotics of its Toeplitz operators; the former actually turns out to coincide with the Hilbert space appearing in the construction of the well-known Barut-Girardello coherent states. Further extension to the case of Legendre polynomials is likewise discussed.

Weighted Shifts on Directed Semi-Trees: an Application to Creation Operators on Segal–Bargmann Spaces

We extend the notion of a weighted shift on a directed tree to the case of a more general graph which we call a directed semi-tree. Some basic properties of such operators are investigated. It is shown that a generalized creation operator on the Segal–Bargman space is unitarily isomorphic to a weighted shift on a directed semi-tree of a particular form.

Toeplitz Operators with Uniformly Continuous Symbols

Let Tf be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi:10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on Cn and the Bergman metric on \({\Omega}\), respectively, the operator Tf is bounded if and only if f is bounded. Moreover, Tf is compact if and only if f vanishes at the boundary of \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).

Heat Flow and An Algebra of Toeplitz Operators

We define a family of associative products $${(\sharp_s)_{s>0}}$$ on a space S ∞ of real analytic functions on $${\mathbb{C}^n}$$ that are contained in the range of the heat transform for all times t > 0. Extending results in Bauer (J Funct Anal 256:3107–3142, 2009), Coburn (J Funct Anal 161:509–525, 1999; Proc Am Math Soc 129(11):3331–3338, 2007) we show that this product leads to composition formulas of in general unbounded Berezin–Toeplitz operators $${T^{(s)}_f}$$ on $${H_s^2}$$ having symbols $${f \in S_{\infty}}$$ . Here $${H_s^2}$$ denotes the Segal–Bargmann space over $${\mathbb{C}^n}$$ with respect to the semi-classical parameter s > 0. In the special case of operators with polynomial symbols or for products of just two operators such formulas previously have been obtained in Bauer (J Funct Anal 256:3107–3142, 2009), Coburn (Proc Am Math Soc 129(11):3331–3338, 2007), respectively. Finally we give an example of a bounded real analytic function h on $${\mathbb{C}}$$ such that $${(T_h^{(1)})^2}$$ cannot be expressed in form of a Toeplitz operator $${T_g^{(1)}}$$ where g fulfills a certain growth condition at infinity.

Paragrassmann algebras as quantum spaces, Part II: Toeplitz Operators

This paper continues the study of paragrassmann algebras begun in Part I with the definition and analysis of Toeplitz operators in the associated holomorphic Segal-Bargmann space. These are defined in the usual way as multiplication by a symbol followed by the projection defined by the reproducing kernel. These are non-trivial examples of spaces with Toeplitz operators whose symbols are not functions and which themselves are not spaces of functions.