Toeplitz operators with BMO symbols on the Segal-Bargmann space

Abstract

Abstract. We show that Zorboska's criterion for compactness of Toeplitz operators with BMO 1 symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on C n . We establish some basic properties of BMO p for p ≥ 1 and complete the characterization of bounded and compact Toeplitz operators with BMO 1 symbols. Via the Bargmann isometry and results of Lo and Englis, we also give a compactness criterion for the Gabor-Daubechies "windowed Fourier localization operators" on L 2 (R n , dv) when the symbol is in a BMO 1 Sobolev-type space. Finally, we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of Toeplitz operators.

Toeplitz Operators from One Fock Space to Another

“…By Lemma 2.3, we know that (3), (4) and (5) are equivalent, moreover, the corresponding norms in (3.4) are equivalent. We now prove (1) implies (5). For each {λ k } ∈ l p , put f as (2.7).…”

“…Toeplitz operators with symbol ϕ ∈ L ∞ , acting on F 2 α , have been well studied. The characterizations on ϕ, for which the induced Toeplitz operators T ϕ is bounded (or compact) on F 2 α , have been considered in [2,3,5,10,[17][18][19]. Recently, Isralowitz and Zhu [11] obtained the boundedness, compactness and Schatten class membership of Toeplitz operators with μ ≥ 0 on the Fock space F 2 α .…”

“…Let 0 < q < p < ∞, and let μ ≥ 0. Set s = p q and s to be the conjugate exponent of s. Then the following statements are equivalent: is a vanishing (p, q)-Fock Carleson measure; IEOT (3) μ t ∈ L s (dv) for some (or any) t > 0; (4) μ(B(·, δ)) ∈ L s (dv) for some (or any) δ > 0; (5) ∞ k=1 μ (B(a k , r)) s < ∞ for some (or any) r > 0.…”

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

“…[2][3][4][5]7,8,14,15]). For example, it was shown in [8] that under some condition on the weight ν the Toeplitz operator T ν g with bounded symbol g is compact on H 2 ν if and only if g ν ∈ C 0 (Ω).…”

“…In case of the Segal-Bargmann space H 2 (C n , dμ t ) of all entire functions square integrable with respect to a t-dependent family of Gaussian measures μ t , (here t > 0 plays the role of the time parameter in the heat flow), it was proved in [3] that for symbols f ∈ BMO 1 (C n ) the Toeplitz operator T t f is bounded (respectively compact) if and only if the heat transform f ( t 2 ) at time t 2 is bounded (respectively vanishing at infinity) (cf. [3,7]). A natural question which arises in the study of Toeplitz operators with a fixed symbol acting on a family of weighted Bergman spaces is whether their compactness is independent of the weight parameter.…”