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Wolfram Bauer

Leibniz Universität Hannover, Faculty of Mathematics and Physics, Faculty Member

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Julita Vassileva

University of Saskatchewan

Knox College

Roshan Chitrakar

Nepal College of Information Technology

Graduate Center of the City University of New York

Adventist University of the Philippines

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RIVERS STATE POLYTECHNIC

Charles University, Prague

Christopher Crick

Oklahoma State University

Margaret Wegener

The University of Queensland, Australia

Roma Tre University, Rome, Italy

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Papers by Wolfram Bauer

Resolvent algebra in Fock-Bargmann representation

arXiv (Cornell University), Aug 13, 2022

The resolvent algebra R(X, σ) associated to a symplectic space (X, σ) was introduced by D. Buchho... more The resolvent algebra R(X, σ) associated to a symplectic space (X, σ) was introduced by D. Buchholz and H. Grundling as a convenient model of the canonical commutation relation (CCR) in quantum mechanics. We first study a representation of R(C n , σ) with the standard symplectic form σ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that R(C n , σ) itself is a Toeplitz algebra. In the sense of R. Werner's correspondence theory we determine its corresponding shift-invariant and closed space of symbols. Finally, we discuss a representation of the resolvent algebra R(H,σ) for an infinite dimensional symplectic separable Hilbert space (H,σ). More precisely, we find a representation of R(H,σ) inside the full Toeplitz algebra over the Fock-Bargmann space in infinitely many variables.

Berezin–Toeplitz quantization and composition formulas

Journal of Functional Analysis, 2009

3331-3338] we derive composition formulas for Berezin-Toeplitz operators with i.g. unbounded symb... more 3331-3338] we derive composition formulas for Berezin-Toeplitz operators with i.g. unbounded symbols in the range of certain integral transforms. The question whether a finite product of Berezin-Toeplitz operators is an operator of this type again can be answered affirmatively in several cases, but there are also well-known counter examples. We explain some consequences of such formulas to C *-algebras generated by Toeplitz operators.

Mean Oscillation and Hankel Operators on the Segal-Bargmann Space

Integral Equations and Operator Theory, 2005

For the Segal-Bargmann space of Gaussian square integrable entire functions on C m we consider Ha... more For the Segal-Bargmann space of Gaussian square integrable entire functions on C m we consider Hankel operators H f with symbols in f ∈ T (C m). We completely characterize the functions in T (C m) for which the operators H f and Hf are simultaneously bounded or compact in terms of the mean oscillation of f. The analogous description holds for the commutators [M f , P ] where M f denotes the "multiplication by f " and P is the Toeplitz projection. These results are already known in case of bounded symmetric domains Ω in C m (see [BBCZ] or [C]). In the present paper we combine some techniques of [BBCZ] and [BC1]. Finally, we characterize the entire function f ∈ H(C m) ∩ T (C m) and the polynomials p in z andz for which the Hankel operators Hf and Hp are bounded (resp. compact).

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

Complex Analysis and Operator Theory

We present a survey and new results on the construction and Gelfand theory of commutative Toeplit... more We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in $$\mathbb {C}^n$$ C n . As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

$Research paper thumbnail of Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces$

Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces

Monatshefte für Mathematik

We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. T... more We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, $$C^{*}$$ C ∗ -algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz structure can be found in the recent literature. In this article, we obtain a Korovkin-type theorem on $$B({\mathcal {H}})$$ B ( H ) which generalizes all such results in the recent literature. As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space $$A^{2}({\mathbb {D}})$$ A 2 ( D ) , Fock space $$F^{2}({\mathbb {C}})$$ F 2 ( C ) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk $${\mathbb {D}}$$ D...

$Research paper thumbnail of Trivializable and Quaternionic Subriemannian Structures on $${\mathbb {S}}^7$$ and Subelliptic Heat Kernel$

Trivializable and Quaternionic Subriemannian Structures on $${\mathbb {S}}^7$$ and Subelliptic Heat Kernel

The Journal of Geometric Analysis

On the seven dimensional Euclidean sphere $${\mathbb {S}}^7$$ S 7 we compare two subriemannian st... more On the seven dimensional Euclidean sphere $${\mathbb {S}}^7$$ S 7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution $${\mathcal {H}}_T$$ H T is induced by a Clifford module structure of $${\mathbb {R}}^8$$ R 8 . More precisely, $${\mathcal {H}}_T$$ H T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution $${\mathcal {H}}_{Q}$$ H Q of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration $${\mathbb {S}}^3\hookrightarrow {\mathbb {S}}^7\rightarrow {\mathbb {S}}^4$$ S 3 ↪ S 7 → S 4 . Answering a question in: Markina and Godoy Molina (Rev Mat Iberoam 27(3), 997–1022, 2011) we first show that $${\mathcal {H}}_{Q}$$ H Q does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures [20], the ...

ON THE HEAT KERNEL ON FORMS ON THE HEISENBERG GROUP (Symmetry and Singularity of Geometric Structures and Differential Equations)

数理解析研究所講究録 = RIMS Kokyuroku, Dec 1, 2019

Self-Adjointness of Toeplitz Operators in the Segal-Bargmann Space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded op... more 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 SegalBargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operatorvalued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.

Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups

Advances in Mathematics, 2020

Pseudo H-type Lie groups G r,s of signature (r, s) are defined via a module action of the Cliffor... more Pseudo H-type Lie groups G r,s of signature (r, s) are defined via a module action of the Clifford algebra Cℓ r,s on a vector space V ∼ = R 2n. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let N r,s denote the Lie algebra corresponding to G r,s. A choice of left-invariant vector fields [X 1 ,. .. , X 2n ] which generate a complement of the center of N r,s gives rise to a second order operator ∆ r,s := X 2 1 +. .. + X 2 n − X 2 n+1 +. .. + X 2 2n , which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of ∆ r,s in the case r = 0, s > 0 and study their properties. In the case of r > 0 we prove that ∆ r,s admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of ∆ r,s and the existence of a fundamental solution in the space of Schwartz distributions.

The fundamental solution of a class of ultra-hyperbolic operators on Pseudo H-type groups

Pseudo H-type Lie groups G_r,s of signature (r,s) are defined via a module action of the Clifford... more Pseudo H-type Lie groups G_r,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓ_r,s on a vector space V R^2n. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let N_r,s denote the Lie algebra corresponding to G_r,s. A choice of left-invariant vector fields [X_1, ..., X_2n] which generate a complement of the center of N_r,s gives rise to a second order operator Δ_r,s:= (X_1^2+ ... + X_n^2)- (X_n+1^2+ ... + X_2n^2 ), which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of Δ_r,s in the case r=0, s>0 and study their properties. In the case of r>0 we prove that Δ_r,s admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of Δ_r,s and the existence of a fundamental solution in the space of Schwartz distributions.

Compactness characterization

For 1 < p < ∞ let T α p be the norm closure of the algebra generated by Toeplitz operators with b... more For 1 < p < ∞ let T α p be the norm closure of the algebra generated by Toeplitz operators with bounded symbols acting on the standard weighted Fock space F p α. In this paper, we will show that an operator A is compact on F p α if and only if A ∈ T α p and the Berezin transform B α (A) of A vanishes at infinity.

Berger-Coburn Theorem, Localized Operators, and the Toeplitz Algebra

Operator Algebras, Toeplitz Operators and Related Topics, 2020

We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators ... more We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators and extend one part of this theorem to the setting of p-Fock spaces (1 ≤ p ≤ ∞). We present an overview of recent results by various authors on the compactness characterization via the Berezin transform for certain operators acting on the Fock space. Based on these results we present three new characterizations of the Toeplitz C * algebra generated by Toeplitz operators with bounded symbols.

Uniform continuity and quantization on bounded symmetric domains

Journal of the London Mathematical Society, 2017

We consider Toeplitz operators T λ f with symbol f acting on the standard weighted Bergman spaces... more We consider Toeplitz operators T λ f with symbol f acting on the standard weighted Bergman spaces over a bounded symmetric domain Ω ⊂ C n. Here λ > genus − 1 is the weight parameter. The classical asymptotic relation for the semi-commutator (*) lim λ→∞ T λ f T λ g − T λ f g = 0, with f, g ∈ C(B n), where Ω = B n denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside Ω (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let β denote the Bergman metric distance function on Ω. We prove that (*) remains true for f and g in the space UC(Ω) of all β-uniformly continuous functions on Ω. Note that this space contains also unbounded functions. In case of the complex unit ball Ω = B n ⊂ C n we show that (*) holds true for bounded symbols in VMO(B n), where the vanishing oscillation inside B n is measured with respect to β. At the same time (*) fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in Ω.

Spectral theory of a class of nilmanifolds attached to Clifford modules

Mathematische Zeitschrift, 2020

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of... more We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-homeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

Toeplitz quantization on Fock space

Journal of Functional Analysis, 2018

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Complex Analysis and Operator Theory, 2018

We study C *-algebras generated by Toeplitz operators acting on the standard weighted Bergman spa... more We study C *-algebras generated by Toeplitz operators acting on the standard weighted Bergman space A 2 λ (B n) over the unit ball B n in C n. The symbols f ac of generating operators are assumed to be of a certain product type, see (1.1). By choosing a and c in different function algebras S a and S c over lower dimensional unit balls B ℓ and B n−ℓ , respectively, and by assuming the invariance of a ∈ S a under some torus action we obtain C *-algebras T λ (S a , S c) whose structural properties can be described. In the case of k-quasi-radial functions S a and bounded uniformly continuous or vanishing oscillation symbols S c we describe the structure of elements from the algebra * This work was partially supported by CONACYT Project 238630, México and by DFG (Deutsche Forschungsgemeinschaft), Project BA 3793/4-1. T λ (S a , S c), derive a list of irreducible representations of T λ (S a , S c), and prove completeness of this list in some cases. Some of these representations originate from a "quantization effect", induced by the representation of A 2 λ (B n) as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.

On the complete integrability of the geodesic flow of pseudo-H-type Lie groups

Analysis and Mathematical Physics, 2018

Pseudo-H-type groups G r ,s form a class of step-two nilpotent Lie groups with a natural pseudo-R... more Pseudo-H-type groups G r ,s form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of G r ,s families of first integrals are constructed. A modification of these functions gives a set of dim G r ,s functionally independent smooth first integrals in involution. The existence of a lattice L in G r ,s is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold L\G r ,s is proved under additional assumptions on the group G r ,s .

Uniformly continuous functions and quantization on the Fock space

Boletín de la Sociedad Matemática Mexicana, 2016

With a family $$(\mu _t)_{t>0}$$(μt)t>0 of Gaussian probability measures we consider the sc... more With a family $$(\mu _t)_{t>0}$$(μt)t>0 of Gaussian probability measures we consider the scale $$(H_t^2)_{>0}$$(Ht2)>0 of $$\mu _t$$μt-square integrable entire functions on $$\mathbb {C}^n$$Cn. Here t plays the role of Planck’s constant. For f and g in the space $$\mathrm{BUC}(\mathbb {C}^n)$$BUC(Cn) of all bounded and uniformly continuous complex valued functions on $$\mathbb {C}^n$$Cn we show the asymptotic composition formula 1$$\begin{aligned} \lim _{t\downarrow 0} \Vert T_f^{(t)} T_g^{(t)} -T^{(t)}_{fg} \Vert _t =0, \end{aligned}$$limt↓0‖Tf(t)Tg(t)-Tfg(t)‖t=0,where $$\Vert \cdot \Vert _t$$‖·‖t denotes the norm in $$\mathcal {L}(H_t^2)$$L(Ht2) and $$T_f^{(t)}$$Tf(t) is the Toeplitz operator with symbol f. Different from previously known results (e.g. Borthwick, Perspectives on quantization. Contemporary mathematics, vol 214. AMS, Providence, pp 23–37, 1998; Coburn, Commun Math Phys 149:415–424, 1992) neither differentiability nor compact support of the operator symbols is assumed. We provide an example which indicates that (1) in general fails for rapidly oscillating bounded symbols.

A codimension 3 sub-Riemannian structure on the Gromoll–Meyer exotic sphere

Differential Geometry and its Applications, 2017

We construct a co-dimension 3 completely non-holonomic sub-bundle on the Gromoll-Meyer exotic 7 s... more We construct a co-dimension 3 completely non-holonomic sub-bundle on the Gromoll-Meyer exotic 7 sphere based on its realization as a base space of a Sp(2)principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a 4n + 3 dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere S 7 .

Sub-Riemannian structures in a principal bundle and their Popp measures

Applicable Analysis, 2017

Abstract In this note, we explain a relation between the Popp measures of sub-Riemannian structur... more

Resolvent algebra in Fock-Bargmann representation

arXiv (Cornell University), Aug 13, 2022

The resolvent algebra R(X, σ) associated to a symplectic space (X, σ) was introduced by D. Buchho... more The resolvent algebra R(X, σ) associated to a symplectic space (X, σ) was introduced by D. Buchholz and H. Grundling as a convenient model of the canonical commutation relation (CCR) in quantum mechanics. We first study a representation of R(C n , σ) with the standard symplectic form σ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that R(C n , σ) itself is a Toeplitz algebra. In the sense of R. Werner's correspondence theory we determine its corresponding shift-invariant and closed space of symbols. Finally, we discuss a representation of the resolvent algebra R(H,σ) for an infinite dimensional symplectic separable Hilbert space (H,σ). More precisely, we find a representation of R(H,σ) inside the full Toeplitz algebra over the Fock-Bargmann space in infinitely many variables.

Berezin–Toeplitz quantization and composition formulas

Journal of Functional Analysis, 2009

3331-3338] we derive composition formulas for Berezin-Toeplitz operators with i.g. unbounded symb... more 3331-3338] we derive composition formulas for Berezin-Toeplitz operators with i.g. unbounded symbols in the range of certain integral transforms. The question whether a finite product of Berezin-Toeplitz operators is an operator of this type again can be answered affirmatively in several cases, but there are also well-known counter examples. We explain some consequences of such formulas to C *-algebras generated by Toeplitz operators.

Mean Oscillation and Hankel Operators on the Segal-Bargmann Space

Integral Equations and Operator Theory, 2005

For the Segal-Bargmann space of Gaussian square integrable entire functions on C m we consider Ha... more For the Segal-Bargmann space of Gaussian square integrable entire functions on C m we consider Hankel operators H f with symbols in f ∈ T (C m). We completely characterize the functions in T (C m) for which the operators H f and Hf are simultaneously bounded or compact in terms of the mean oscillation of f. The analogous description holds for the commutators [M f , P ] where M f denotes the "multiplication by f " and P is the Toeplitz projection. These results are already known in case of bounded symmetric domains Ω in C m (see [BBCZ] or [C]). In the present paper we combine some techniques of [BBCZ] and [BC1]. Finally, we characterize the entire function f ∈ H(C m) ∩ T (C m) and the polynomials p in z andz for which the Hankel operators Hf and Hp are bounded (resp. compact).

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

Complex Analysis and Operator Theory

We present a survey and new results on the construction and Gelfand theory of commutative Toeplit... more We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in $$\mathbb {C}^n$$ C n . As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

$Research paper thumbnail of Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces$

Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces

Monatshefte für Mathematik

We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. T... more We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, $$C^{*}$$ C ∗ -algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz structure can be found in the recent literature. In this article, we obtain a Korovkin-type theorem on $$B({\mathcal {H}})$$ B ( H ) which generalizes all such results in the recent literature. As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space $$A^{2}({\mathbb {D}})$$ A 2 ( D ) , Fock space $$F^{2}({\mathbb {C}})$$ F 2 ( C ) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk $${\mathbb {D}}$$ D...

$Research paper thumbnail of Trivializable and Quaternionic Subriemannian Structures on $${\mathbb {S}}^7$$ and Subelliptic Heat Kernel$

Trivializable and Quaternionic Subriemannian Structures on $${\mathbb {S}}^7$$ and Subelliptic Heat Kernel

The Journal of Geometric Analysis

On the seven dimensional Euclidean sphere $${\mathbb {S}}^7$$ S 7 we compare two subriemannian st... more On the seven dimensional Euclidean sphere $${\mathbb {S}}^7$$ S 7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution $${\mathcal {H}}_T$$ H T is induced by a Clifford module structure of $${\mathbb {R}}^8$$ R 8 . More precisely, $${\mathcal {H}}_T$$ H T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution $${\mathcal {H}}_{Q}$$ H Q of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration $${\mathbb {S}}^3\hookrightarrow {\mathbb {S}}^7\rightarrow {\mathbb {S}}^4$$ S 3 ↪ S 7 → S 4 . Answering a question in: Markina and Godoy Molina (Rev Mat Iberoam 27(3), 997–1022, 2011) we first show that $${\mathcal {H}}_{Q}$$ H Q does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures [20], the ...

ON THE HEAT KERNEL ON FORMS ON THE HEISENBERG GROUP (Symmetry and Singularity of Geometric Structures and Differential Equations)

数理解析研究所講究録 = RIMS Kokyuroku, Dec 1, 2019

Self-Adjointness of Toeplitz Operators in the Segal-Bargmann Space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded op... more 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 SegalBargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operatorvalued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.

Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups

Advances in Mathematics, 2020

Pseudo H-type Lie groups G r,s of signature (r, s) are defined via a module action of the Cliffor... more Pseudo H-type Lie groups G r,s of signature (r, s) are defined via a module action of the Clifford algebra Cℓ r,s on a vector space V ∼ = R 2n. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let N r,s denote the Lie algebra corresponding to G r,s. A choice of left-invariant vector fields [X 1 ,. .. , X 2n ] which generate a complement of the center of N r,s gives rise to a second order operator ∆ r,s := X 2 1 +. .. + X 2 n − X 2 n+1 +. .. + X 2 2n , which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of ∆ r,s in the case r = 0, s > 0 and study their properties. In the case of r > 0 we prove that ∆ r,s admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of ∆ r,s and the existence of a fundamental solution in the space of Schwartz distributions.

The fundamental solution of a class of ultra-hyperbolic operators on Pseudo H-type groups

Pseudo H-type Lie groups G_r,s of signature (r,s) are defined via a module action of the Clifford... more Pseudo H-type Lie groups G_r,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓ_r,s on a vector space V R^2n. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let N_r,s denote the Lie algebra corresponding to G_r,s. A choice of left-invariant vector fields [X_1, ..., X_2n] which generate a complement of the center of N_r,s gives rise to a second order operator Δ_r,s:= (X_1^2+ ... + X_n^2)- (X_n+1^2+ ... + X_2n^2 ), which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of Δ_r,s in the case r=0, s>0 and study their properties. In the case of r>0 we prove that Δ_r,s admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of Δ_r,s and the existence of a fundamental solution in the space of Schwartz distributions.

Compactness characterization

For 1 < p < ∞ let T α p be the norm closure of the algebra generated by Toeplitz operators with b... more For 1 < p < ∞ let T α p be the norm closure of the algebra generated by Toeplitz operators with bounded symbols acting on the standard weighted Fock space F p α. In this paper, we will show that an operator A is compact on F p α if and only if A ∈ T α p and the Berezin transform B α (A) of A vanishes at infinity.

Berger-Coburn Theorem, Localized Operators, and the Toeplitz Algebra

Operator Algebras, Toeplitz Operators and Related Topics, 2020

We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators ... more We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators and extend one part of this theorem to the setting of p-Fock spaces (1 ≤ p ≤ ∞). We present an overview of recent results by various authors on the compactness characterization via the Berezin transform for certain operators acting on the Fock space. Based on these results we present three new characterizations of the Toeplitz C * algebra generated by Toeplitz operators with bounded symbols.

Uniform continuity and quantization on bounded symmetric domains

Journal of the London Mathematical Society, 2017

We consider Toeplitz operators T λ f with symbol f acting on the standard weighted Bergman spaces... more We consider Toeplitz operators T λ f with symbol f acting on the standard weighted Bergman spaces over a bounded symmetric domain Ω ⊂ C n. Here λ > genus − 1 is the weight parameter. The classical asymptotic relation for the semi-commutator (*) lim λ→∞ T λ f T λ g − T λ f g = 0, with f, g ∈ C(B n), where Ω = B n denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside Ω (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let β denote the Bergman metric distance function on Ω. We prove that (*) remains true for f and g in the space UC(Ω) of all β-uniformly continuous functions on Ω. Note that this space contains also unbounded functions. In case of the complex unit ball Ω = B n ⊂ C n we show that (*) holds true for bounded symbols in VMO(B n), where the vanishing oscillation inside B n is measured with respect to β. At the same time (*) fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in Ω.

Spectral theory of a class of nilmanifolds attached to Clifford modules

Mathematische Zeitschrift, 2020

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of... more We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-homeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

Toeplitz quantization on Fock space

Journal of Functional Analysis, 2018

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Complex Analysis and Operator Theory, 2018

We study C *-algebras generated by Toeplitz operators acting on the standard weighted Bergman spa... more We study C *-algebras generated by Toeplitz operators acting on the standard weighted Bergman space A 2 λ (B n) over the unit ball B n in C n. The symbols f ac of generating operators are assumed to be of a certain product type, see (1.1). By choosing a and c in different function algebras S a and S c over lower dimensional unit balls B ℓ and B n−ℓ , respectively, and by assuming the invariance of a ∈ S a under some torus action we obtain C *-algebras T λ (S a , S c) whose structural properties can be described. In the case of k-quasi-radial functions S a and bounded uniformly continuous or vanishing oscillation symbols S c we describe the structure of elements from the algebra * This work was partially supported by CONACYT Project 238630, México and by DFG (Deutsche Forschungsgemeinschaft), Project BA 3793/4-1. T λ (S a , S c), derive a list of irreducible representations of T λ (S a , S c), and prove completeness of this list in some cases. Some of these representations originate from a "quantization effect", induced by the representation of A 2 λ (B n) as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.

On the complete integrability of the geodesic flow of pseudo-H-type Lie groups

Analysis and Mathematical Physics, 2018

Pseudo-H-type groups G r ,s form a class of step-two nilpotent Lie groups with a natural pseudo-R... more Pseudo-H-type groups G r ,s form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of G r ,s families of first integrals are constructed. A modification of these functions gives a set of dim G r ,s functionally independent smooth first integrals in involution. The existence of a lattice L in G r ,s is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold L\G r ,s is proved under additional assumptions on the group G r ,s .

Uniformly continuous functions and quantization on the Fock space

Boletín de la Sociedad Matemática Mexicana, 2016

With a family $$(\mu _t)_{t>0}$$(μt)t>0 of Gaussian probability measures we consider the sc... more With a family $$(\mu _t)_{t>0}$$(μt)t>0 of Gaussian probability measures we consider the scale $$(H_t^2)_{>0}$$(Ht2)>0 of $$\mu _t$$μt-square integrable entire functions on $$\mathbb {C}^n$$Cn. Here t plays the role of Planck’s constant. For f and g in the space $$\mathrm{BUC}(\mathbb {C}^n)$$BUC(Cn) of all bounded and uniformly continuous complex valued functions on $$\mathbb {C}^n$$Cn we show the asymptotic composition formula 1$$\begin{aligned} \lim _{t\downarrow 0} \Vert T_f^{(t)} T_g^{(t)} -T^{(t)}_{fg} \Vert _t =0, \end{aligned}$$limt↓0‖Tf(t)Tg(t)-Tfg(t)‖t=0,where $$\Vert \cdot \Vert _t$$‖·‖t denotes the norm in $$\mathcal {L}(H_t^2)$$L(Ht2) and $$T_f^{(t)}$$Tf(t) is the Toeplitz operator with symbol f. Different from previously known results (e.g. Borthwick, Perspectives on quantization. Contemporary mathematics, vol 214. AMS, Providence, pp 23–37, 1998; Coburn, Commun Math Phys 149:415–424, 1992) neither differentiability nor compact support of the operator symbols is assumed. We provide an example which indicates that (1) in general fails for rapidly oscillating bounded symbols.

A codimension 3 sub-Riemannian structure on the Gromoll–Meyer exotic sphere

Differential Geometry and its Applications, 2017

We construct a co-dimension 3 completely non-holonomic sub-bundle on the Gromoll-Meyer exotic 7 s... more We construct a co-dimension 3 completely non-holonomic sub-bundle on the Gromoll-Meyer exotic 7 sphere based on its realization as a base space of a Sp(2)principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a 4n + 3 dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere S 7 .

Sub-Riemannian structures in a principal bundle and their Popp measures

Applicable Analysis, 2017

Abstract In this note, we explain a relation between the Popp measures of sub-Riemannian structur... more