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Self-commutators of Toeplitz operators and isoperimetric inequalities

Steven R. Bell, Timothy Ferguson and Erik Lundberg
Mathematical Proceedings of the Royal Irish Academy
Vol. 114A, No. 2 (2014), pp. 115-132
Published by: Royal Irish Academy
DOI: 10.3318/pria.2014.114.03
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Page Count: 18
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Self-commutators of Toeplitz operators and isoperimetric inequalities
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Abstract For a hyponormal operator, C.R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.

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