Algebras Of Toeplitz Operators Research Articles

Spectral theorem approach to commutative C⁎-algebras generated by Toeplitz operators on the unit ball: Quasi-elliptic related cases

We consider commutative C⁎-algebras of Toeplitz operators in the weighted Bergman space on the unit ball in Cn. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.

Toeplitz Operators on CR Manifolds and Group Actions

Let X be a connected orientable compact CR manifold with non-degenerate Levi form. In this paper, we study the algebra of Toeplitz operators on X and we establish star product for a certain class of symbols on X. In the second part of this paper, we consider a locally free action of a compact Lie group G on X and we investigate the associated algebra of G-invariant Toeplitz operators. The main application is a construction of deformation quantization of compact “quantizable” pseudo-Kähler manifolds.

Commutative Algebras of Toeplitz Operators on the Bergman Space Revisited: Spectral Theorem Approach

For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols.

Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

In this paper, we construct a family of Berezin–Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${\\mathcal O}(p^{3/4})$ around the points $p\\Lambda$, where $\\Lambda$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigenvalues from the cluster.We show that it provides a Berezin–Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.

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.

On asymptotic expansions of generalized Bergman kernels on symplectic manifolds

A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner Laplacian is constructed.

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

We study $C^*$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ over the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type. By choosing $a$ and $c$ in different function algebras $\mathcal{S}_a$ and $\mathcal{S}_c$ over lower dimensional unit balls $\mathbb{B}^{\ell}$ and $\mathbb{B}^{n-\ell}$, respectively, and by assuming the invariance of $a\in \mathcal{S}_a$ under some torus action we obtain $C^*$-algebras $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$ whose structural properties can be described. In the case of $k$-quasi-radial functions $\mathcal{S}_a$ and bounded uniformly continuous or vanishing oscillation symbols $\mathcal{S}_c$ we describe the structure of elements from the algebra $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, derive a list of irreducible representations of $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, and prove completeness of this list in some cases. Some of these representations originate from a `quantization effect', induced by the representation of $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ 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.

Examples of infinite direct sums of spectral triples

We study two ways of summing an infinite family of noncommutative spectral triples. First, we propose a definition of the integration of spectral triples and give an example using algebras of Toeplitz operators acting on weighted Bergman spaces over the unit ball of $\mathbb{C}^n$. Secondly, we construct a spectral triple associated to a general polygonal self-similar set in $\mathbb{C}$ using algebras of Toeplitz operators on Hardy spaces. In this case, we show that we can recover the Hausdorff dimension of the fractal set.

Quasi-parabolic Composition Operators on Weighted Bergman Spaces

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_{\alpha}^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $\varphi(z)=$ $z+\psi(z)$, where $\psi\in$ $H^{\infty}(\mathbb{H})$ and $\Im(\psi(z)) > \epsilon > 0$. We especially examine the case where $\psi$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

Spectral triples and Toeplitz operators

We give examples of spectral triples, in the sense of A. Connes, constructed using the algebra of Toeplitz operators on smoothly bounded strictly pseudoconvex domains in C, or the star product for the Berezin–Toeplitz quantization. Our main tool is the theory of generalized Toeplitz operators on the boundary of such domains, due to Boutet de Monvel and Guillemin. VERSION – 2.1, February 11, 2014 1Research supported by GA CR grant no. 201/12/G028. 2Mathematics Institute, Silesian University in Opava, Na Rybnicku 1, 74601 Opava, Czech Republic, and Mathematics Institute, Academy of Sciences, Žitna 25, 11567 Prague 1, Czech Republic 3Aix-Marseille Universite, CNRS, CPT, UMR 7332, 13288 Marseille France, et Universite de Toulon

CommutativeC⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.

Commutative algebras of Toeplitz operators on the strip (in Russian)

Commutative algebras of Toeplitz operators on the strip (in Russian)

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.

Toeplitz Operators with Quasi-Homogeneuos Quasi-Radial Symbols on Some Weakly Pseudoconvex Domains

On the weakly pseudo-convex domains \(\Omega _p^n\) we introduce quasi-homogeneous quasi-radial symbols. These are used to prove the existence of a commutative Banach algebra of Toeplitz operators on Bergman space of \(\Omega _p^n\). We also show that group theoretic and geometric properties for our symbols are satisfied. The results presented here contain the geometric description of the symbols introduced by Vasilevski in (Integr Equ Operat Theory, 66(1):141–152, 2010) for the unit ball \(\mathbb {B}^n\).

English

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existence of a strong link between such symbols and algebras with the geometry of $\mathbb{P}^n(\mathbb{C})$. The latter is also proved for the corresponding symbols and algebras on $\mathbb{B}^n$.

On C*-Algebras of Toeplitz Operators on the Harmonic Bergman Space

Commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk have been completely classified in terms of geometric properties of the symbol class. The question when two Toeplitz operators acting on the harmonic Bergman space commute is still open. In some papers, conditions on the symbols have been given in order to have commutativity of two Toeplitz operators. In this paper, we describe three different algebras of Toeplitz operators acting on the harmonic Bergman space: The C*-algebra generated by Toeplitz operators with radial symbols, in the elliptic case; the C*-algebra generated by Toeplitz operators with piecewise continuous symbols, in the parabolic and hyperbolic cases. We prove that the Calkin algebra of the first two algebras are commutative, like in the case of the Bergman space, while the last one is not.

On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

The stability of the Nystrom method for the Muskhelishvili equation on piecewise smooth simple contours Γ is studied. It is shown that in the space L 2 the method is stable if and only if certain operators Aτj from an algebra of Toeplitz operators are invertible. The operators Aτj depend on the parameters of the equation considered, on the opening angles θj of the corner points τj ∈ Γ, and on parameters of the approximation method mentioned. Numerical experiments show that there are opening angles where the operators Aτj are noninvertible. Therefore, for contours with such corners the method under consideration is not stable. Otherwise, the method is always stable. Numerical examples show an excellent convergence of the method.

Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours

AbstractThe stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points cj, j = 0, 1, …, m relies on the invertibility of certain operators belonging to an algebra of Toeplitz operators. The operators do not depend on the shape of the contour, but on the opening angle θj of the corresponding corner cj and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle In the interval (0.1π, 1.9π), it is found that there are 8 values of θj where the invertibility of the operator may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.

Commutative C*-Algebras of Toeplitz Operators on Complex Projective Spaces

We prove the existence of commutative C*-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space \({{\mathbb{P}^n}\mathbb{(C)}}\). The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of \({{\mathbb{P}^n}\mathbb{(C)}}\).

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H2 with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where ψ∈H∞(H) and ℑ(ψ(z))>ϵ>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.