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

摘要

For Banach space operators T satisfying the Tadmor-Ritt condition parallel to(zI - T)(-1)parallel toless than or equal toC - 1(-1), z > 1, we prove that the best-possible constant C-T(n) bounding the polynomial calculus for T, parallel top(T)parallel to less than or equal to C-T(n)parallel topparallel to(infinity), deg(p) less than or equal to n, behaves (in the worst case) as log n as n --> infinity. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree. (C) 2003 Elsevier Inc. All rights reserved.