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A Theorem of Brown–Halmos Type for Bergman Space Toeplitz Operators

✍ Scribed by Patrick Ahern; Željko Čučković

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
146 KB
Volume
187
Category
Article
ISSN
0022-1236

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✦ Synopsis

We study the analogues of the Brown-Halmos theorem for Toeplitz operators on the Bergman space. We show that for f and g harmonic, T f T g =T h only in the trivial case, provided that h is of class C 2 with the invariant laplacian bounded.

Here the trivial cases are f ¯or g holomorphic. From this we conclude that the zeroproduct problem for harmonic symbols has only the trivial solution. Finally, we provide examples that show that the Brown-Halmos theorem fails for general symbols, even for symbols continuous up to the boundary.

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A Dichotomy for Linear Spaces of Toeplit
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✍ Edward A Azoff; Marek Ptak 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 293 KB

Let S be a linear manifold of bounded Hilbert space operators. An operator A belongs to the reflexive closure of S if Af belongs to the closure of S f for each vector f in the underlying Hilbert space. Two extreme possibilities are (1) S is reflexive in the sense that ref S=S, and (2) S is transitiv