Functional calculus under Kreiss type conditions

It is shown that for an algebraic Banach space operator T , the Kreiss condition, ‖( zI – T ) –1 ‖ ≤ $ {C \over {| z | - 1} } $ , | z | > 1, implies the following functional calculus estimate$$ \Vert f (T) \Vert \le {16 \over {\pi} }\, C \cdot {\rm deg} (T) \, \Vert f \Vert _{\infty}\, , $$where deg( T ) is the degree of the minimal polynomial annihilating T . This result extends the known estimates of the powers of T for Kreiss operators on finite dimensional spaces. In the case of a general Kreiss operator, an estimate of the rational calculus is proved:$$ \Vert r(T) \Vert \le {16 \over {\pi} }\, C ( {\rm deg}(r) + 1) \, \Vert r \Vert _{\infty} \, . $$Similar estimates hold for the polynomial calculus under generalized Kreiss conditions. A link is also established between the sharp constant in the first estimate and the norm of the best solution for a Nevanlinna–Pick type interpolation problem in analytic Besov classes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)