2004
DOI: 10.1090/s0002-9939-04-07362-9
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Hankel operators with antiholomorphic symbols on the Fock space

Abstract: Abstract. We consider Hankel operators of the form H z k : F m := {f : f is entire and C n |f (z)| 2 e −|z| m < ∞} → L 2 (e −|z| m ). Here k, m, n ∈ N. We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if m > 2k. PreliminariesThe investigation of Hankel operators on the Bergman space of certain domains Ω has a long history. See, for example, [24] and [25]. Furthermore, there have been some attempts to characterize compactness of Hankel operators on L 2 s… Show more

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Cited by 19 publications
(29 citation statements)
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“…We remember the analogous situation for the classic Hankel operator H z k (for details see the introduction; the mentioned results for the classic Hankel operator can be found in [19]): These Hankel operators are not bounded. However, using the projection onto A 2, 1 C, |z| 2 instead of onto A 2 C, |z| 2 will not lead to an "improvement" of operator theoretic properties in the cases k > 2.…”
Section: Theorem 45mentioning
confidence: 95%
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“…We remember the analogous situation for the classic Hankel operator H z k (for details see the introduction; the mentioned results for the classic Hankel operator can be found in [19]): These Hankel operators are not bounded. However, using the projection onto A 2, 1 C, |z| 2 instead of onto A 2 C, |z| 2 will not lead to an "improvement" of operator theoretic properties in the cases k > 2.…”
Section: Theorem 45mentioning
confidence: 95%
“…Remark 1.4 It should be added that for m ≥ 2k the Hankel operator H z k : A 2 (C, |z| m ) → L 2 (C, |z| m ) is bounded, but for m < 2k it is not bounded. (See [19].) Remark 1.5 Note that it is possible to investigate Hankel operators with symbols in a wide class of antiholomorphic symbols with the methodology of [19].…”
Section: Theorem 13 For M > 2k the Hankel Operatormentioning
confidence: 99%
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“…See also [16] and [21] for the one-dimensional case. It would be of interest to characterize Stieltjes moment sequences having this property.…”
Section: Proofmentioning
confidence: 99%