Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

Abstract

The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L 2 -spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as L 2 -spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic functions, and so on. Both positive and negative results are obtained.

Berezin transform and Toeplitz operators on harmonic Bergman spaces

“…(using the fact that the kernel is real) as is noticed in [10]; the integrand remains unchanged upon interchanging u and v. Thus the Berezin transform of the commutator T u T v − T v T u is always identically zero, but T u T v = T v T u is quite possible. This also suggests that the harmonic analogue of the Axler-Zheng theorem would not be true in general, because T u T v − T v T u might probably not always be compact.…”

“…They also thank the referee both for pointing out an error in the earlier version of the paper with helpful suggestions and for drawing their attention to the reference [10].…”

Berezin transform on the harmonic Fock space

“…Of course, the proportionality of norms (12) implies that the same holds also for the corresponding inner products. Using the decomposition (10), we thus have…”

“…One such candidate, namely, the pluriharmonic Bergman spaces L 2 ph , consisting of all functions f in L 2 for which ∂ 2 f /∂ z j ∂ z k = 0 ∀ j, k, has recently been investigated in [12] and [13]. Unfortunately, it turned out that the analogue of (4),…”

Berezin Quantization of Gaussian Functions Depending by a Quantum and Compression Parameter

“…In the first two sections we will review basic notions on the Berezin quantization of C n and the quantum harmonic oscillator, references are [1], [2], [3], [4], [5], [6] and [7]. The other sections are devoted to the proofs of the following results.…”