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Functional calculus under Kreiss type conditions
Author(s) -
Vitse Pascale
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310341
Subject(s) - mathematics , functional calculus , type (biology) , banach space , polynomial , norm (philosophy) , calculus (dental) , operator (biology) , holomorphic functional calculus , pure mathematics , space (punctuation) , algebraic number , algebra over a field , mathematical analysis , approximation property , computer science , law , medicine , ecology , biochemistry , chemistry , dentistry , repressor , political science , transcription factor , gene , biology , operating system
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)