Related papers: Berezin-Toeplitz quantization revisited

In this paper we prove that if the polar decomposition of a symbol $f$ is truncated above, i.e., $f(re^{i\theta} )=\sum_{k=-\infty}^Ne^{ik\theta} f_k (r)$ where the $f_k$'s are radial functions, and if the associated Toeplitz operator $T_f$…

This paper mainly studies totally Abelian operators in the context of analytic Toeplitz operators on both the Hardy and Bergman space. When the symbol is a meromorphic function on $\mathbb{C}$, we establish the connection between totally…

We investigate the Berezin-Toeplitz operators that operate on the geometric quantized space corresponding to the $SO(3)$-Witten-Chern-Simons theory. We conjecture that the $SO(3)$-Berezin-Toeplitz operators quantized from the A-polynomial…

In recent years, the near diagonal asymptotics of the equivariant components of the Szeg\"{o} kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of…

This is a survey paper. We discuss Toeplitz operators in K\"ahler geometry, with applications to geometric quantization, and review some recent developments.

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both,…

Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde \X$ its Oshima compactification, and $(\pi,\mathrm{C}(\widetilde \X))$ the regular representation of $G$ on $\widetilde \X$. We study integral operators on…

Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group $SU_q(2)$ is such…

This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate…

We prove that Toeplitz operators associated with a Bernstein-Markov measure on a compact complex manifold endowed with a big line bundle form an algebra under composition. As an application, we derive a Szeg\H{o}-type spectral…

Let $\mathscr{T}(L^{\infty}(\mathbb{T}))$ be the Toeplitz algebra, that is, the $C^*$-algebra generated by the set $\{T_{\phi} : \phi\in L^{\infty}(\mathbb{T})\}$. Douglas's theorem on symbol map states that there exists a $C^*$-algebra…

In this paper we investigate the possibility of constructing a complete quantization procedure consisting of geometric and deformation quantization. The latter assigns a noncommutative algebra to a symplectic manifold, by deforming the…

We demonstrate that the structure of complex second-order strongly elliptic operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$,…

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $\chi_k(T_P)$ the functional calculus of $T_P$ by…

We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we…

We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers,…

We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can…

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal…

We study invertibility and compactness of positive Toeplitz operators associated to a continuous Parseval frame on a Hilbert space. As applications, we characterize compactness of affine and Weyl-Heisenberg localization operators as well as…

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras…