International Journal of Modern Science and Technology · ISSN 2456-0235

Research Article · Biotechnology & Bioeng. · Volume 2, Issue 2 · Feb 2017 · Pages 56–60

On Numerical Ranges of Convexoid Operators

J. A. Otieno, N. B. Okelo, O. Ongati

  • J. A. Otieno: School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O Box 210-40601, Bondo-Kenya
  • N. B. Okelo: School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O Box 210-40601, Bondo-Kenya
  • O. Ongati: School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O Box 210-40601, Bondo-Kenya

Abstract

Numerical range is useful in studying operators on Hilbert spaces. In particular, the geometrical properties of numerical range often provide useful information about algebraic and analytic properties of an operator. The theory of numerical range played a crucial role in the study of some algebraic structures especially in the non-associative context. The numerical range of an operator depends strongly upon the base field. Motivated by theoretical study and applications, researchers have considered different generalizations of numerical range. Numerical range of an operator may be a point, or a line segment containing none, one or all of its end points. Numerical range of another operator may be an open set, closed set or neither. In this paper, we give results of numerical range of convexoid operators. Let be an infinite dimensional complex Hilbert space and be algebra of all bounded linear operators on . is said to be convexoid if the closure of the numerical range coincides with the convex hull of its spectrum. In this paper, we determine the numerical ranges of convexoid operators. We employ some results for convexoid operators due to Furuta and numerical ranges due to Shapiro, Furuta and Nakamoto, Mecheri and Okelo. Some properties of numerical ranges are also discussed.

Keywords

Numerical range; Convexoid operator; Numerical radius; Spectrum

References

  1. Bauer FL. On the field of values subordinate to a norm . Numer Math . 1962;4:103-113.
  2. Furuta T. Some characteristics of convexoid o perators. Rev Roum Maths Press et Appl. 1973;18:893-900.
  3. Furuta T, Nakamoto R. On the numerical range of an operator. Proc Japan Acad Soc. 1971;47:279-284.
  4. Halmos PR. A Hilbert space problem book. Von Nostrand Princeton, 1967.
  5. Lumer G. Semi inner product space. Trans Am Math Soc. 1961;100:29-43.
  6. Mecheri S. The nume rical range of linear operators. Filomat. 2008;2:1-8.
  7. Okelo NB, Kisengo SK, Bonyo JO. Properties of Hilbert space operators . Operator Theory. Lambert Academic Publishing, 2013.
  8. Seddik A. The numerical r ange of elementary operators II . Linear Algebra Appl. 2001;338:239-244.
  9. Seddik A. On numerical range a nd norm of elementary operators. linear Multilinear Algebra. 2004;52:293-302.
  10. Shapiro JH. Notes on the nume rical range. Michigan S tate University, East Lansing, 2004.
  11. Toeplitz O. Das algebraische Analogon zu einem Satz von Feje’r. Math. Z. 1969;2:187-197.
  12. Vijayabalaji S, Shyamsundar G. Interval-valued intuition istic fuzzy transition matrices. International Journal of Modern Science and Technology. 2016;1(2):47-51.
  13. Judith J O, Okelo NB, Roy K, Onyango T. Numerical Solutions of Mathematical Model on Effects of Biological Control on Cereal Aphid Population Dynamics. International Journal of Modern Science and Technology. 2016;1(4)138-143.
  14. Judith J O, Ok elo NB, Roy K, Onyango T. Construction and Qualitative Analysis of Mathematical Model for Biological Control on Cereal Aphid Population Dynamics . International Journal of Modern Science and Technology. 2016;1(5)150-158.
  15. Vijayabalaji S, Sathiyaseelan N. Interval- Valued Product Fuzzy Soft Matrices and its Application in Decision Making, International Journal of Modern Science and Technology. 2016;1(6)159-163.
  16. Chinnadurai V, Bharathivelan K. Cubic Ideals in Near Subtraction Semigroups, International Journa l of Modern Science and Technology. 2016;1(8)276-282.