Abstract: For a relatively compact measurable subset $E$ of the unit ball in $n$-dimensional space, we consider the Toeplitz operator $T_E$
on the Bergman space whose symbol is the indicator function of the set $E$. We explain how this operator is related to basic phenomena of complex analysis such as Runge approximation and Hartogs extension. Thanks to the biholomorphic invariance of Bergman spaces,
it is possible to represent the spectrum of such an operator using complex-hyperbolic integral invariants of the set $E$. Using such representations, we prove some eigenvalue inequalities for the operator $T_E$, reminiscent of the Faber-Krahn inequality for the Dirichlet Laplacian.
Mathematics Seminars and Colloquia