Self-commutators of Toeplitz operators and isoperimetric inequalities
arXiv:1211.6937 · doi:10.3318/PRIA.2014.114.03
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.
15 pages, 2 figures. Now published in Mathematical Proceedings of the Royal Irish Academy. This version differs slightly in apperance from the published one. Changes from Previous Version: Fixed small error in the statements and proofs of Theorems 1.1 and 1.2 and Corollary 3.1. Other minor changes from last version