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Singly generated algebras containing a compact operator

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Recent Advances in Operator Theory and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 127))

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Abstract

For an operator T, consider P (T), the norm closed algebra generated by T and the identity operator. The question is, “When does this algebra contain a compact operator?”. The question remains unanswered. However it is shown that the set of all operators such that P (T) contains a compact operator is dense in \(\mathcal{B}(\mathcal{H})\) and the interior of this set is characterized. If the term “compact operator” is replaced by “finite rank operator” or if P (T) is replaced by the weakly closed or weak-star closed algebra generated by T and the identity, the same results are obtained. Similar questions are raised and answered for other algebras associated with an operator.

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References

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Author information

Authors and Affiliations

  1. University of Tennessee, Knoxville, TN, 37996-1300, USA

    John B. Conway

  2. Bucknell University, Lewisburg, PA, 17837, USA

    Gabriel Prăjitură

Authors
  1. John B. Conway
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  2. Gabriel Prăjitură
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Editor information

Editors and Affiliations

  1. University of Szeged, Aradi Vértanuúk Tere 1, 6720, Szeged, Hungary

    László Kérchy

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    I. Gohberg

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47405-4301, USA

    Ciprian I. Foias

  4. Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8-10/1411, 1040, Wien, Austria

    H. Langer

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© 2001 Springer Basel AG

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Conway, J.B., Prăjitură, G. (2001). Singly generated algebras containing a compact operator. In: Kérchy, L., Gohberg, I., Foias, C.I., Langer, H. (eds) Recent Advances in Operator Theory and Related Topics. Operator Theory: Advances and Applications, vol 127. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8374-0_9

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  • DOI: https://doi.org/10.1007/978-3-0348-8374-0_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9539-2

  • Online ISBN: 978-3-0348-8374-0

  • eBook Packages: Springer Book Archive

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