Let Ω be a bounded convex Reinhardt domain in C 2 and φ ∈ C(Ω). We show that the Hankel operator H φ is compact if and only if φ is holomorphic along every non-trivial analytic disc in the boundary of Ω.Let Ω be a domain in C n and let L 2 (Ω) and A 2 (Ω) denote square integrable functions on Ω and the Bergman space on Ω (the set of square integrable holomorphic functions on Ω), respectively. Since A 2 (Ω) is a closed subspace in L 2 (Ω) the Bergman projection P : L 2 (Ω) → A 2 (Ω), the orthogonal projection, exists. Furthermore, let H φ f = (I − P)(φ f ) for all f ∈ A 2 (Ω) and φ ∈ L ∞ (Ω). We note that H φ is called the Hankel operator with symbol φ. We refer the reader to [Pel03, Zhu07] and references there in for more information on these operators.Hankel operators form an active research area in operator theory. Our interest lies in their compactness properties in relation to the behavior of the symbols on the boundary of the domain. On the unit disc D in C Axler ([Axl86]) showed that, for f holomorphic on the unit disc D, the Hankel operator H f is compact on A 2 (D) if and only if f is in the little Bloch space (that is, (1 − |z| 2 )| f (z)| → 0 as |z| → 1). This result has been extended into higher dimensions by Peloso ([Pel94]) in case the domain is smooth bounded and strongly pseudoconvex. The same year, Li ([Li94]) characterized bounded and compact Hankel operators on strongly pseudoconvex domains for symbols that are square integrable only. Recently,Čučković and the second author [ČŞ09, Theorem 3] gave a characterization for compactness of Hankel operators on smooth bounded convex domains in C 2 with symbols smooth up to the boundary. We note that even though they stated their result for smooth domains and smooth symbols on the closure, examination of the proof shows that C 1 -smoothness of the domain and the symbol is sufficient. They proved the following theorem.Theorem (Čučković-Şahutoglu). Let Ω be a C 1 -smooth bounded convex domain in C 2 and φ ∈ C 1 (Ω). Then the Hankel operator H φ is compact on A 2 (Ω) if and only if φ • f is holomorphic for any holomorphic function f : D → bΩ.In this paper we prove a similar result with symbols that are only continuous up to the boundary. The first result in this direction was proven by Le in [Le10]. He showed that for Ω = D n ,
Let Ω be a bounded pseudoconvex domain in C 2 with Lipschitz boundary or a bounded convex domain in C n and φ ∈ C(Ω) such that the Hankel operator H φ is compact on the Bergman space A 2 (Ω). Then φ • f is holomorphic for any holomorphic f : D → bΩ.Let Ω be a domain in C n and A 2 (Ω) denote the Bergman space of Ω, the space of square integrable holomorphic functions on Ω. Since A 2 (Ω) is a closed subspace of L 2 (Ω), the space of square integrable functions on Ω, there exists an orthogonal projection P :Hankel operators have been well studied on the Bergman space of the unit disc. Sheldon Axler in [Axl86] proved the following interesting theorem.The space of holomorphic functions satisfying the condition in the theorem is called little Bloch space. One can check that φ(z) = exp((z + 1)/(z − 1)) is bounded on D but it does not belong to the little Bloch space. Hence not every bounded symbol that is smooth on the domain produces compact Hankel operator on the disc. However, Hankel operators with symbols continuous on the closure are compact for bounded domains in C (see, for instance, [Şah12, Proposition 1]). We refer the reader to [Zhu07] for more information on the theory of Hankel operators (as well as Toeplitz operators) on the Bergman space of the unit disc. We note that Sheldon Axler's result has been extended to a small class of domains in C n , such as strongly pseudoconvex domains, by Marco Peloso [Pel94] and Huiping Li [Li94].The situation in C n for n ≥ 2 is radically different. For instance, H z 1 is not compact when Ω is the bidisc (see, for instance, [Le10, Clo17b, CŞ18, Clo17a]). Hence in higher dimensions compactness of Hankel operators is not guaranteed even if the symbol is smooth up to the boundary. We refer the reader to [Str10, Has14] for more information about Hankel operators in higher dimensions and their relations to ∂-Neumann problem.
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