Linear operators on polynomials preserving roots in open circular domains

Abstract:

In the present paper we answer a question raised by J. Borcea and P. Brändén and give a description of the class of operators preserving roots in open circular domains, i.e., in images of the open upper half-plane under the Möbius transformations. Our second result is a description of the difference between $\mathcal A(G)$ (the class of operators preserving roots in an open set $G$) and $\mathcal A(\overline G)$ (the class of operators preserving roots in $\overline {G}$).