ISSN 2456-0235

INDEXED IN 

​​​​​​International Journal of Modern Science and Technology, Vol. 2, No. 9, 2017, Pages 318-321. 


Spectral Characterization of Convexoid Operators  

N. B. Okelo*, J. A. Otieno, O. Ongati
School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O Box 210-40601, Bondo-Kenya.
​​*Corresponding author’s e-mail: bnyaare@yahoo.com

Abstract
Spectrum is a very useful property in studying operators on Banach spaces particularly, Hilbert spaces. In particular, the geometrical properties of spectrum often provide useful information about algebraic and analytic properties of an operator. The theory of spectrum played a crucial role in the study of some algebraic structures especially in the associative context. The spectrum of an operator depends strongly upon the base of scalars.  Motivated by theoretical study and applications, researchers have considered different generalizations of spectrum. This study gives results of spectrum of convexoid operators. Let H be an infinite dimensional complex Hilbert space and B(H) be algebra of all bounded linear operators on H. T ϵ B(H) is said to be convexoid if the closure of the numerical range coincides with the convex hull of its spectrum. In this paper, the study determines the spectrum of convexoid operators. This work considers some results on spectra due to Rota, Hildebrandant, Furuta and Nakamoto among others.

Keywords: Resolvent set; Convexoid operator; Spectral radius; Spectrum of operator. 

References

  1. Bauer FL.   On the field of values subordinate to a norm. Numer Math. 2015; 4:103-113.
  2. Furuta T.  Some characteristics of convexoid operators. Rev Roum Maths Press et Appl. 2013;18:893-900.
  3. Furuta T,  Nakamoto R. On the numerical range of an operator. Proc Japan Acad Soc. 2010;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. 2011;100:29-43.
  6. Mecheri  S.  The numerical 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. Otieno JA, Okelo NB, Ongati O. On Numerical Ranges of Convexoid Operators. International Journal of Modern Science and Technology. 2017;2(2):56-60.
  9. Seddik A.  The numerical range of elementary operators II. Linear Algebra Appl. 2011;338:239-244.
  10. Seddik A.  On numerical range and norm of elementary operators, linear Multilinear Algebra, 52, (2014), 293-302.
  11. Shapiro JH.  Notes on the numerical range, Michigan State University, East Lansing, 2004.
  12. Toeplitz O. Das algebraische Analogon zu einem Satz von Feje’r, Math Z. 2009;2:187-197.
  13. Vijayabalaji S, Shyamsundar G.  Interval-valued intuitionistic fuzzy transition matrices. International Journal of Modern Science and Technology. 2016;1(2)47-51.
  14.  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​​.
  15.  Judith J O,  Okelo 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​​.
  16.  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(7):159-163​​.
  17.  Chinnadurai V,  Bharathivelan K. Cubic Ideals in Near Subtraction Semigroups. International Journal of Modern Science and Technology. 2016;1(8):276-282​​. 

International Journal of Modern Science and Technology