Contractivity of the H-calculus and Blaschke Products

  • Christoph Kriegler
  • Lutz Weis
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 195)

Abstract

It is well known that a densely defined operator A on a Hilbert space is accretive if and only if A has a contractive H-calculus for any angle bigger than \( \tfrac{\pi } {2} \). A third equivalent condition is that \( \left\| {\left( {A - w} \right)\left( {A + \bar w} \right)^{ - 1} } \right\| \leqslant 1 \) for all Re w≥0. In the Banach space setting, accretivity does not imply the boundedness of the H-calculus any more. However, we show in this note that the last condition is still equivalent to the contractivity of the H-calculus in all Banach spaces. Furthermore, we give a sufficient condition for the contractivity of the H-calculus on ℂ+, thereby extending a Hilbert space result of Sz.-Nagy and Foias to the Banach space setting.

Mathematics Subject Classification (2000)

47 A60 47B44 30D50 

Keywords

H-calculus accretive operators Blaschke products 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Christoph Kriegler
    • 1
  • Lutz Weis
    • 1
  1. 1.Institut für AnalysisUniversität KarlsruheKarlsruheGermany

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