Study recovers C*-algebra from fields of Toeplitz algebras on specific groups.
problem Recovering C*-algebra from fields of Toeplitz algebras on specific groups.
method Using continuous fields of Toeplitz algebras and a crossed product.
result Algebra of principal symbols can be recovered from fields of Toeplitz algebras.
Researchers compute spectral flow of Toeplitz operators on manifolds and domains.
problem Computing spectral flow for families of Toeplitz operators.
method Using Callias-type operators and cohomological formulas, the spectral flow is related to the index of these operators.
result A cohomological formula for spectral flow on even-dimensional manifolds.
Study of spectral flow in symmetric Toeplitz operator families.
problem Understanding spectral flow in families of symmetric Toeplitz operators.
method Analog of Atiyah-Singer-Robbin-Salamon theorem for Z2-valued spectral flow. result Graded secondary spectral flow equals secondary index of a Callias-type operator.
Study rigidity theorems for odd dimensional manifolds using Toeplitz operators.
problem Rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds.
method Combining modular method, modular transgression, and analysis of odd Chern classes.
result Fundamental group of manifolds is crucial in odd dimensions for rigidity.
Study optimal holomorphic extensions on complex manifolds with transitivity property.
problem Optimal holomorphic extensions on complex manifolds with transitivity property.
method Use Toeplitz operators and transitivity property for optimal holomorphic extensions.
result Transitivity property of optimal holomorphic extensions with small defect.
The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.
problem Quantization of compact symplectic manifolds with higher Landau levels.
method Develops Berezin-Toeplitz quantization using a Bochner Laplacian with specific spectral properties.
result The quantization provides a formal star-product for the lowest Landau level.
Study on spectral asymptotics of Toeplitz operators on CR manifolds.
problem Analyzing spectral properties of Toeplitz operators on CR manifolds.
method Full asymptotic expansion of functional calculus of Toeplitz operators.
result Established several CR analogues of complex geometry results.
New findings on optimization landscape of Toeplitz covariance estimation.
problem Understanding the geometry of the Gaussian maximum-likelihood objective for Toeplitz covariance estimation.
method Overparameterized Carathéodory representation of positive definite Toeplitz covariance matrices, focusing on both amplitudes and frequencies.
result Joint optimization of amplitudes and frequencies leads to a benign population landscape, allowing for global recovery of the true Toeplitz covariance.
The paper studies quantization on symplectic manifolds with real polarizations, comparing different quantization methods.
problem Quantization on compact symplectic manifolds with real polarizations.
method Geometric quantization, Toeplitz operators, Fourier transforms, asymptotic expansion of traces.
result Deformation quantization is realized through asymptotic traces of Toeplitz operators.
The goal is to understand the index-theoretic aspects of the recent preprint of R. Nest and F. Radulescu, math.OA/9911042. The basic observation (due to E. Guenter/N. Higson) is that the index of the Toeplitz operator is equal to the index of an associated Callias type operator, i.e. a Dirac operator with potential, th…
We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator. In the proof of this paper, we use some properties of a circle and use Higson's a…
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.
Develops quantization for non-compact complex manifolds with spectral gap.
problem Quantization of non-compact complex manifolds with spectral gap.
method Berezin-Toeplitz quantization, spectral gap analysis, asymptotic expansion.
result Toeplitz operators form a closed algebra and satisfy a complete composition expansion.
Boundary conditions added to elliptic complexes on manifolds.
problem Adding boundary conditions to elliptic complexes.
method Developed Fredholm theory for complexes of Toeplitz type pseudodifferential operators.
result Boundary conditions without projections can be chosen if a topological obstruction vanishes.
We establish an index theorem for Toeplitz operators on odd dimensional spin manifolds with boundary. It may be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to…
New star-product defined on Poisson manifolds using Toeplitz operators.
problem Defining star-products on Poisson manifolds induced by symplectic Lie algebroids.
method Using Toeplitz operators on groupoids with Heisenberg group structure.
result Generalization of Guillemin and Melrose's symplectic approach.
Quantizes symplectic manifolds with toric singularities using Toeplitz operators.
problem Quantize symplectic manifolds with toric singularities.
method Establishes quantization for compact toric symplectic manifolds with transversal singular real polarizations using Toeplitz operators.
result Toeplitz operators determine a star product on compact toric symplectic manifolds with toric singularities as ℏo0+. We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for no…
Study small eigenvalues of Toeplitz operators and their relation to Mabuchi geodesics.
problem Analyzing small eigenvalues of Toeplitz operators on complex projective manifolds.
method Proving the existence of exponentially decaying eigenvalues for Toeplitz operators with specific symbols, and establishing a connection to Mabuchi geodesics.
result Logarithmic distribution of small eigenvalues correlates with Mabuchi geodesics between polarizations.
We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application…
Constructs families of Toeplitz operators for symplectic fibrations.
problem Quantization of symplectic fibrations.
method Smooth families of Szegö projections and Toeplitz operators.
result Deformation quantization of prequantizable symplectic fibrations.
Formula for Toeplitz operator kernel on CR manifolds.
problem Analyzing Toeplitz operators on CR manifolds.
method Formula for the symbol of the kernel, asymptotic expansions.
result Formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel.
We prove an off-diagonal expansion for a Toeplitz operator with an indicator function.
problem Asymptotics of Toeplitz operators with indicator function
method Off-diagonal expansion
result We extend two results to the non-compact setting.
We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.
Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.
problem Quantization of symplectic manifolds with bounded geometry.
method Berezin-Toeplitz quantization theory.
result Correct semiclassical limit achieved.
Paper studies spectral measures of semi-classical Toeplitz operators.
problem Understanding spectral properties of semi-classical Toeplitz operators.
method Asymptotic expansion in powers of ℏ for spectral measure μℏ. result Asymptotic expansion of spectral measure for semi-classical Toeplitz operators.
Improved Moser-Trudinger-Onofri inequality with constraints on sphere.
problem Improving constant in Moser-Trudinger-Onofri inequality with constraints.
method Generalization to higher order moments and perturbation method.
result New inequalities with similarity to Lebedev-Milin type inequalities.
Study eigenvalues of Toeplitz operators on symplectic manifolds with discrete wells.
problem Understanding eigenvalues of Toeplitz operators on symplectic manifolds.
method Semiclassical analysis of Toeplitz operators on high tensor powers of a positive line bundle.
result Upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit.
Geometric properties of Toeplitz kernels relate to circle function injectivity.
problem Injectivity of Toeplitz operators on the unit circle.
method Linking Toeplitz operators to geodesics in Grassmann manifolds.
result Existence of geodesics in Grassmann manifolds connects Toeplitz operator injectivity.
Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.
problem Analyzing the asymptotics of weighted Bergman kernels for submultiplicative filtrations.
method Demonstrated that weight operator is a Toeplitz operator; analyzed asymptotics of weighted Bergman kernels.
result Local refinement of convergence of jumping measures towards geodesic ray pushforward measure.
Paper extends index theorem to odd-dimensional manifolds with even-dimensional boundaries.
problem Index theorem for odd-dimensional manifolds with boundaries.
method Equivariant Toeplitz index theory.
result Established equivariant version of Dai-Zhang's theorem.
Solves problem of describing transformations for upper triangular Toeplitz operators.
problem Describing coordinate transformations preserving upper triangular Toeplitz form of operator fields.
method Implicit formulas involving matrix-valued functions for describing transformations and Nijenhuis operators.
result Formulas for coordinate transformations and Nijenhuis operators in upper triangular Toeplitz form.
The Laplace operator is approximated using Berezin-Toeplitz quantization.
problem Approximating the Laplace operator on Hodge manifolds.
method Self-adjoint operator on holomorphic sections approximates the Laplace operator via Berezin-Toeplitz quantization.
result The approximation error tends to zero with higher polarization powers.
We obtain the semi-classical expansion of the kernels and traces of Toeplitz operators with $\cC^k$--\,symbol on a symplectic manifold. We also give a semi-classical estimate of the distance of a Toeplitz operator to the space of self-adjoint and multiplication operators.
We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin-Toeplitz quantization obtained recently by Ma-Marinescu, and of the composition of two Berezin-Toeplitz quantizations. Our main tool is the stationary phase formula of Melin-Sjöstrand.
Isomorphic algebra connects Toeplitz to Heisenberg group.
problem Connecting Toeplitz algebra to Heisenberg group.
method Isomorphism between Toeplitz algebra and Heisenberg group ideal.
result Found isomorphism between algebra and Heisenberg group ideal.
Calculates a key coefficient for symplectic manifold operators.
problem Computing the composition of Berezin-Toeplitz operators on symplectic manifolds.
method Using the full-off diagonal expansion of the Bergman kernel.
result Computed the second coefficient of the composition.
For phase-space manifolds which are compact Kaehler manifolds relations between the Berezin-Toeplitz quantization and the quantization with the help of Berezin's coherent states and symbols are studied. First the results on the Berezin-Toeplitz quantization of arbitrary compact Kaehler manifolds due to Bordemann, Meinr…
A new model uses Toeplitz matrices to analyze time-series data transitions.
problem Analyzing transitions in time-series data from nonautonomous systems.
method Deep Koopman-layered models with learnable Toeplitz matrices, leveraging Toeplitz matrices' universal property.
result The model demonstrates universality and generalization, outperforming existing methods.
New tools for analyzing Kähler manifolds, proving operator algebra and asymptotic kernel.
problem Analyzing Berezin-Toeplitz operators on Kähler manifolds.
method Introducing new tools for analytic microlocal analysis.
result Space of analytic Berezin-Toeplitz operators is an algebra.
Study quantizes eigenstates of Bochner-Laplacian on symplectic manifolds.
problem Quantizing eigenstates of the Bochner-Laplacian on symplectic manifolds.
method Using eigenstates of the renormalized Bochner Laplacian, we apply Berezin-Toeplitz quantization.
result The quantization has correct semiclassical behavior and a corresponding star-product is constructed.
Quantum propagation studied for Berezin-Toeplitz operators.
problem Asymptotic behavior of quantum propagators and spectral projectors.
method Geometric analysis of Hamiltonian flows and Maslov indices.
result Introduction of quantum states associated with Lagrangian submanifolds.
Geometric quantization for symplectic maps via Toeplitz operators.
problem Quantization of symplectic maps and Witten's conjecture.
method Berezin-Toeplitz operators and holomorphic sections over Kähler manifolds.
result Established a semi-classical trace formula for quantum representations of mapping class groups.
Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin-Toeplitz quantization of the open set of M where the curvature on L is non-degenerate. The quantum spaces are the spectral spaces corresponding to [0,k−N] (N>1 fixed), of the Kodaira…
Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operator…
Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.
Let M be a complete Riemannian manifold and assume that M is partitioned by a hypersurface N. In this paper we introduce a novel class of functions Cw(M) on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of φ that belongs to Cw(M) we construc…
We establish some subprincipal estimates for Berezin-Toeplitz operators on symplectic compact manifolds. From this, we construct a family of subprincipal symbol maps and we prove that these maps are the only ones satisfying some expected conditions.