Characterizing invertibility of a Toeplitz operator Tφ on the Bergman space in terms of the
properties of the symbol φ has been a long standing open problem. Partial answers can be given
by assuming the positivity of the symbol. In this talk we introduce a novel geometric condition on
the image of the symbol φ, which relaxes the standard positivity condition. We will also discuss the
related Douglas’ problem. (Joint work with Jari Taskinen)