Norm Estimates for the Commutator of a Toeplitz Operator: Timothy Ferguson, Steve Bell, and I have recently obtained (preprint) lower estimates for the norm of the self-commutator of a Toeplitz operator with analytic symbol acting on Bergman space. This followed a similar study of D. Khavinson (1985) in Smirnov Space.

An attractive feature of the D. Khavinson's study was that the lower estimate when combined with C.R. Putnam's inequality (giving an upper estimate) recovers the classical isoperimetric inequality (with sharp constants) relating area and perimeter of the underlying domain.

Putnam's inequality is true in general for hyponormal operators. Thus, we decided to attempt a lower estimate within a setting different from Smirnov Space. Our lower bound for Bergman space involves the first Dirichlet eigenvalue for the Laplacian of the underlying domain (i.e., fundamental frequency or base tone of a drum), and when combined with Putnam's inequality we recover a different isoperimetric inequality: the classical Faber-Krahn inequality. However, in this case the constants are not sharp. We give an alternate estimate that recovers the Saint-Venant inequality for torsional rigidity (again with a non-sharp constant). This led to a conjecture regarding the deficiency in the constants.

Open Problem: We conjecture that our main result is sharp, and that Putnam's inequality which is sharp in general is not in the restricted setting of our paper. An improved Putnam's inequality in this setting could (combined with one of our results) recover the Saint-Venant inequality with sharp constants.

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