Articles

VARIOUS NOTIONS OF ORTHOGONALITY IN NORMED SPACES

  • N.B. OKELO ,
  • J.O. AGURE ,
  • P.O. OLECHE
Expand
  • School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, Box 210-40601, Bondo, Kenya; School of Mathematics, Statistics and Actuarial Science, Maseno University, Box 333, Maseno, Kenya

Received date: 2011-09-27

  Revised date: 2012-12-12

  Online published: 2013-09-20

Supported by

The first author´s appreciations go to National Council for Science and Technology (NCST) of Kenya and DAAD- Germany for the financial support to this research.

Abstract

In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid-ered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.

Cite this article

N.B. OKELO , J.O. AGURE , P.O. OLECHE . VARIOUS NOTIONS OF ORTHOGONALITY IN NORMED SPACES[J]. Acta mathematica scientia, Series B, 2013 , 33(5) : 1387 -1397 . DOI: 10.1016/S0252-9602(13)60090-9

References

[1] Anderson J. On normal derivations. Proc Amer Math Soc, 1973, 38: 135–140

[2] Bachir A, Segres A. Numerical range and orthogonality in normed spaces. Filmat, 2009, 23: 21–41

[3] Bachman G, Narici L. Functional Analysis. New York: Academic Press, 2000

[4] Benitez C. Orthogonality in normed linear spaces: a classification of the different concepts and some open problems. Revista Mathematica, 1989, 2: 53–57

[5] Bhattacharya D K, Maity A K. Semilinear tensor product of ??-Banach algebras. Ganita, 1989, 40(2): 75–80

[6] Bonsall F, Duncan J. Complete Normed Algebra. New York: Springer-Verlag, 1973

[7] Bonyo J O, Agure J O. Norm of a derivation and hyponormal operators. Int J Math Anal, 2010, 4(14): 687–693

[8] Bouali S,Bouhafsi Y. On the range of the elementary operator X 7→ AXA − X. Math Proc Royal Irish Academy, 2008, 108: 1–6

[9] Cabrera M, Rodriguez A. Nondegenerately ultraprime Jordan Banach algebras. Proc London Math Soc, 1994, 69: 576–604

[10] Canavati J A, Djordjevic S V, Duggal B P. On the range closure of an elementary operator. Bull Korean Math Soc, 2006, 43: 671–677

[11] Du H K, Ji G X. Norm attainability of elementary operators and derivations. Northeast Math J, 1994, 3: 394–400

[12] Du H K, Wang Y Q, Gao G B. Norms of elementary operators. Proc Amer Math Soc, 2008, 4: 1337–1348

[13] Dutta T K, Nath H K, Kalita R C. -derivations and their norms in projective tensor products of ??-Banach algebras. J London Math Soc, 1998, 2(2): 359–368

[14] Franka M B. Tensor products of C-algebras, operator spaces and Hilbert C-modules. Math Comm, 1999, 4: 257–268

[15] Gajendragadkar P. Norm of derivations of von-Neumann algebra. J Trans Amer Math Soc, 1972, 170: 165–170

[16] Gohberg I C, Krein M G. Introduction to the Theory of Linear Nonselfadjoint Operators. Transl Math Monogr, Vol 18. Providence, RI: Amer Math Soc, 1969

[17] Helemskii A. Lectures and exercises on Functional Analysis. Translation of Mathematical Monographs, Vol 233. New York: American Mathematical Society, 2006

[18] Iain R, Dana P W. Equivalence and Continuous-Trace C-Algebras. New York: American Mathematical Society, 1998

[19] Kadison R, Lance C, Ringrose J. Derivations and automorphisms of operator algebra II. Math J Funct Anal, 1967, 1: 204–221

[20] Keckic D J. Orthogonality in C1 and C1 spaces and normal derivations. J Operator Theory, 2004, 51: 89–104

[21] Keckic D J. Orthogonality of the range and kernel of some elementary operators. Proc Amer Math Soc, 2008, 11: 3369–3377

[22] Kittaneh F. Normal derivations in normal ideals. Proc Amer Math Soc, 1995, 6: 1979–1985

[23] Kreyzig E. Introductory Functional Analysis with Applications. New York: John Wiley and Sons, 1978

[24] Kyle J. Norms of derivations. J London Math Soc, 1997, 16 297–312

[25] Magajna B. The norm of a symmetric elementary operator. Proc Amer Math Soc, 2004, 132: 1747–1754

[26] Mathieu M. Properties of the product of two derivations of a C-algebra. Canad Math Bull, 1990, 42: 115–120

[27] Mathieu M. More properties of the product of two derivations of a C-algebra. Canad Math Bull, 1990, 42: 115–120

[28] Mathieu M. Elementary operators on Calkin algebras. Irish Math Soc Bull, 2001, 46: 33–42

[29] Masamichi T. Theory of Operator Algebras I. New York: Springer-Verlag, 1979

[30] Mecheri S. On the range and kernel of the elementary operators nPi=1AiXBi − X. Acta Math Univ Com-nianae, 2003, 52: 119–126

[31] Mecheri S. Finite operators. Demonstratio Math, 2002, 35: 355–366

[32] Murphy J G. C-algebras and Operator Theory. London: Academic Press Inc, Oval Road, 1990

[33] Turnsek A. Orthogonality in Cp classes. Monatsh Math, 2001, 132: 349–354

[34] Williams J P. Finite operators. Proc Amer Math Soc, 1970, 26: 129–135

Outlines

/