Functional calculus under the Tadmor–Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor–Ritt condition ||(zI−T)−1||⩽C|z−1|−1, |z|>1, we prove that the best-possible constant CT(n) bounding the polynomial calculus for T, ||p(T)||⩽CT(n)||p||∞, deg(p)⩽n, behaves (in the worst case) as log n as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree