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Archiv der Mathematik

, Volume 99, Issue 5, pp 467–479 | Cite as

Generation and commutation properties of the Volterra operator

  • A. F. M. ter Elst
  • Manfred Sauter
  • Jaroslav Zemánek
Article

Abstract

Let V be the classical Volterra operator on L 2(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert–Schmidt operators and as well in the space \({\mathcal{B}(L_2(0,1))}\) equipped with the weak operator topology. Moreover, the algebra generated by V 2 and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.

Mathematics Subject Classification (2010)

Primary 47B38 Secondary 46L99 47L75 47L05 

Keywords

Volterra operator Self-commutator Commutant Trace classoperator Hilbert–Schmidt operator 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • A. F. M. ter Elst
    • 1
  • Manfred Sauter
    • 1
  • Jaroslav Zemánek
    • 2
  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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