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)


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 


H-calculus accretive operators Blaschke products 


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  1. [ADM] D. Albrecht, X. Duong, A. McIntosh, Operator theory and harmonic analysis. Proc. Centre Math. Appl. Austral. Nat. Univ. 34(pt.3), 77–136 (1996).MathSciNetGoogle Scholar
  2. [CDMY] M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach space operators with a bounded H functional calculus. J. Austral. Math. Soc., Ser. A 60, No.1, 51–89 (1996).MATHCrossRefMathSciNetGoogle Scholar
  3. [Dal] E.B. Davies, One parameter semigroups. London Mathematical Society, Monographs, No. 15, 230 p. (1980).Google Scholar
  4. [Da2] E.B. Davies, Non-unitary scattering and capture. I: Hilbert space theory. Comm. Math. Phys. 71, 277–288 (1980).MATHCrossRefMathSciNetGoogle Scholar
  5. [Dru] S. Drury, Remarks on von Neumann’s inequality. Proc. Spec. Year Analysis, Univ. Conn. 1980–81, Lecture Notes in Math. 995, 12–32 (1983).Google Scholar
  6. [Foi] C. Foiaş, Sur certains théorèmes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math. Szeged 18, 15–20 (1957).MATHMathSciNetGoogle Scholar
  7. [Gar] J.B. Garnett, Bounded analytic functions. Revised 1st ed. Graduate Texts in Mathematics 236. Springer-Verlag. xiv, 460 p. (2006).Google Scholar
  8. [KW] P.C. Kunstmann, L. Weis, Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus. Lecture Notes in Math. 1855, 65–311 (2004).Google Scholar
  9. [LM] C. Le Merdy, H -functional calculus and applications to maximal regularity. Publ. Math. UFR Sci. Tech. Besançon. 16, 41–77 (1998).Google Scholar
  10. [NF] B. Sz.-Nagy, C. Foiaş, Harmonic analysis of operators on Hilbert space. North-Holland Publishing Co. xiii, 387 p. (1970).Google Scholar
  11. [vN] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951).MATHMathSciNetGoogle Scholar
  12. [Roo] P.G. Rooney, Laplace transforms and generalized Laguerre polynomials. Canad. J. Math. 10, 177–182 (1958).MATHMathSciNetGoogle Scholar
  13. [Sho] J. Shohat, Laguerre polynomials and the Laplace transform. Duke Math. J. 6, 615–626 (1940).CrossRefMathSciNetGoogle Scholar
  14. [WW] L. Weis, D. Werner, The Daugavet equation for operators not fixing a copy of C [0, 1]. J. Operator Theory 39, No. 1, 89–98 (1998).MATHMathSciNetGoogle Scholar

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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