Extremal domains for self-commutators in the Bergman space. (English) Zbl 1317.47028
Summary: In J.-F. Olsen and M. C. Reguera [“On a sharp estimate for Hankel operators and Putnam’s inequality”, Preprint, arXiv:1305.5193], it was shown that Putnam’s inequality for the norm of self-commutators can be improved by a factor of \(\frac{1}{2}\) for Toeplitz operators with analytic symbol \(\varphi \) acting on the Bergman space \(A^{2}(\Omega)\). This improved upper bound is sharp when \(\varphi (\Omega)\) is a disk.
In this paper, we show that disks are the only domains for which the upper bound is attained.
In this paper, we show that disks are the only domains for which the upper bound is attained.
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47A63 | Linear operator inequalities |
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 30H20 | Bergman spaces and Fock spaces |
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