Area differences under analytic maps and operators

Abstract

Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping h and that of zh, we study various L² norms for T_ϕ(h), where T_ϕ is the Toeplitz operator with symbol ϕ. In Theorem 2.1, given polynomials p and q we find a symbol ϕ such that T_ϕ(p) = q. We extend some of our results to the polydisc.