Toeplitz Operators on Generalized Bergman Spaces

Abstract

We consider the weighted Bergman spaces HL 2 (B d , µ λ ), where we set dµ λ (z) = c λ (1 − |z| 2 ) λ dτ (z), with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert-Schmidt operators on the generalized Bergman spaces.Mathematics Subject Classification (2010). Primary 47B35; Secondary 32A36, 81S10.

Analytic Continuation of Toeplitz Operators

“…We finally remark that the main idea of the proof of the last theorem, namely, integrating by parts with respect to the operators D and D, clearly resembles some of those in Chailuek and Hall [13] (cf., e.g., Theorem 9 there). Somewhat related techniques in the context of bounded symmetric domains appear in Arazy and Upmeier [2], so one wonders whether results analogous to those from this section could not be established also on bounded symmetric domains .…”

“…Also, clearly R(z) * = R(z). Recall that = K * K is an elliptic DO on ∂ of order −1, and, by (13) for α = 0, T is positive selfadjoint. By the property (P1) of generalized Toeplitz operators, there exists an elliptic ϒ ∈ −1 such that ϒ = ϒ = ; we can actually take ϒ to be positive selfadjoint as well (see p. 636 in [15]).…”

“…The methods for proving most of the results about T (α,ρ) φ are more elementary and much closer in spirit to [13].…”

“…However, it may still happen that T (α) φ f , for f in some dense subset (e.g., for f a polynomial), is given by an expression which depends holomorphically on α and extends by analyticity to the range α > −n − 1; so one then has a (densely defined) operator T (α) φ even for these α, which in the sense just described is an "analytic continuation" of the operator (6). The simplest example is of course T (α) 1 = I , which makes sense for all α. Toeplitz operators on B n of this kind were considered by Chailuek and Hall [13], who gave, among others, a variety of alternative definitions of T (α) φ , studied their properties (showing, e.g., that (7) and (8) no longer hold in general) and established boundedness criteria. For φ holomorphic, T (α) φ f = φ f is still just the operator of "multiplication by φ" for any α; the boundedness of these in the extended range −2 < α < −1 on the unit disc D = B 1 in C was studied by Stegenga [21], who gave (rather complicated) criteria involving logarithmic capacity.…”

Construção geométrica de "star-product" integral em espaços simpléticos simétricos não compactos

“…Por outro lado, para garantir que o espaço HL Caso contrario, quando α > −1, este espaçoé não trivial (para mais detalhes pode se ver [7]). …”