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Ranges of vector states on irreducible operator semigroups

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Abstract

Let \(\varphi \) be a linear functional of rank one acting on an irreducible semigroup \(\mathcal {S}\) of operators on a finite- or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of \(\varphi \) cannot be a singleton. We start a study of possible finite ranges for such functionals. In particular, we prove that in certain cases, the existence of a single such functional \(\varphi \) with a two-element range yields valuable information on the structure of \(\mathcal {S}\).

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Acknowledgments

L.W. Marcoux was research supported in part by NSERC (Canada). M. Omladič was research supported in part by the Slovenian Research Agency – ARRS (Slovenia).

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    L. W. Marcoux & H. Radjavi

  2. Faculty of Mathemtics, Natural Sciences, and Information Technologies, University of Primorska, Glagoljaška 8, SI-6000, Koper, Slovenia

    M. Omladič

  3. School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

    A. I. Popov

  4. Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, 49138-15759, Iran

    B. R. Yahaghi

Authors
  1. L. W. Marcoux
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  2. M. Omladič
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  3. A. I. Popov
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  4. H. Radjavi
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  5. B. R. Yahaghi
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Corresponding author

Correspondence to M. Omladič.

Additional information

Communicated by Anthony Lau.

This paper was submitted while the fifth name author was on sabbatical leave at the University of Waterloo. He would like to thank Department of Pure Mathematics of the University of Waterloo, and in particular L.W. Marcoux and H. Radjavi, for their support.

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Marcoux, L.W., Omladič, M., Popov, A.I. et al. Ranges of vector states on irreducible operator semigroups. Semigroup Forum 93, 264–304 (2016). https://doi.org/10.1007/s00233-015-9772-7

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  • DOI: https://doi.org/10.1007/s00233-015-9772-7

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