Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators

Bernard Okelo

doi:10.33434/cams.442556

Araştırma Makalesi

Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators

Yıl 2018, , 91 - 98, 24.12.2018

Bernard Okelo

https://doi.org/10.33434/cams.442556

Öz

Various aspects of elementary operators have been characterized by many mathematicians. In this paper, we consider norm-attainability and orthogonality of these operators in Banach spaces. Characterizations and generalizations of norm-attainability and orthogonality are given in details. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in Banach spaces.

Anahtar Kelimeler

Range-Kernel orthogonality, Elementary operator

Kaynakça

[1] C. Benitez, Orthogonality in normed linear spaces: a classification of the different concepts and some open problems, Rev. Mat., 2 (1989), 53-57.
[2] J. Anderson, On normal derivations, Proc. Amer. Math. Soc., 38 (1973), 135-140.
[3] F. Kittaneh, Normal derivations in normal ideals, Proc. Amer. Math. Soc., 6 (1995), 1979-1985.
[4] S. Mecheri, On the range and kernel of the elementary operators $\sum_{i=1}^{n}A_{i}XB_{i}-X$}, Acta Math. Univ. Comnianae, 52 (2003), 119-126.
[5] A. Bachir, A. Segres, Numerical range and orthogonality in normed spaces, Filomat, 23 (2009), 21-41.
[6] G. Bachman, L. Narici, Functional Analysis, Academic press, New York, 2000.
[7] D. K. Bhattacharya, A. K. Maity, Semilinear tensor product of $\Gamma$- Banach algebras, Ganita, 40(2) (1989), 75-80.
[8] F. Bonsall, J. Duncan, Complete Normed Algebra, Springer Verlag, New York, 1973.
[9] J. O. Bonyo, J. O. Agure, Norm of a derivation and hyponormal operators, Int. J. Math. Anal., 4(14) (2010), 687-693.
[10] S. Bouali, Y. Bouhafsi, On the range of the elementary operator $X\mapsto AXA-X$, Math. Proc. R. Ir. Acad., 108 (2008), 1-6.
[11] M. Cabrera, A. Rodriguez, Nondegenerately ultraprime Jordan Banach algebras, Proc. London Math. Soc., 69 (1994), 576-604.
[12] J. A. Canavati, S. V. Djordjevic, B. P. Duggal, On the range closure of an elementary operator, Bull. Korean Math. Soc., 43 (2006), 671-677.
[13] H. K. Du, G. X. Ji, Norm attainability of elementary operators and derivations, Northeast. Math. J., 3 (1994), 394-400.
[14] H. K. Du, Y. Q. Wang, G. B. Gao, Norms of elementary operators, Proc. Amer. Math. Soc., 4 (2008), 1337-1348.
[15] T. K. Dutta, H. K. Nath, R. C. Kalita, $\alpha$-derivations and their norms in projective tensor products of $\Gamma$-Banach algebras, J. London Math. Soc., 2(2) (1998), 359-368.
[16] M. B. Franka, Tensor products of C*-algebras, operator spaces and Hilbert C*-modules, Math. Comm., 4 (1999), 257-268.
[17] P. Gajendragadkar, Norm of derivations of Von-Neumann algebra, J. Trans. Amer. Math. Soc., 170 (1972), 165-170.
[18] R. Kadison, C. Lance, J. Ringrose, Derivations and automorphisms of operator algebra II, Math. J. Funct. Anal., 1 (1967), 204-221.
[19] D. J. Keckic, Orthogonality in $C_{1}$ and $C_{\infty}$ spaces and normal derivations, J. Operator Theory, 51 (2004), 89-104.
[20] D. J. Keckic, Orthogonality of the range and kernel of some elementary operators, Proc. Amer. Math. Soc., 11 (2008), 3369-3377.
[21] E. Kreyzig, Introductory Functional Analysis with Applications, John Wiley and Sons, New York, 1978.
[22] J. Kyle, Norms of derivations, J. London Math. Soc., 16 (1997), 297-312.
[23] B. Magajna, The norm of a symmetric elementary operator, Proc. Amer. Math., Soc. 132 (2004), 1747-1754.
[24] M. Mathieu, Properties of the product of two derivations of a C*- algebra, Canad. Math. Bull., 42 (1990), 115-120.
[25] M. Mathieu, More properties of the product of two derivations of a C*- algebra, Canad. Math. Bull., 42 (1990), 115-120.
[26] M. Mathieu, Elementary operators on Calkin algebras, Irish Math. Soc. Bull., 46 (2001), 33-42.
[27] J. G. Murphy, C*-algebras and Operator Theory, Academic Press Inc., Oval Road, London, 1990.
[28] N. B. Okelo, J. O. Agure, D. O. Ambogo, Norms of elementary operators and characterization of norm - attainable operators, Int. J. Math. Anal., 24 (2010), 687-693.
[29] N. B. Okelo, J. O. Agure, P. O. Oleche, Various notions of orthogonalty in normed spaces, Acta Math. Sci., 33B(5) (2013) 1387–1397.
[30] A. Turnsek, Orthogonality in $\mathcal{C}_{p}$ classes, Monatsh. Math., 132 (2001), 349 - 354.
[31] J. P. Williams, Finite operators, Proc. Amer. Math. Soc., 26 (1970), 129- 135.

Yıl 2018, , 91 - 98, 24.12.2018

Bernard Okelo

https://doi.org/10.33434/cams.442556

Öz

Kaynakça

[1] C. Benitez, Orthogonality in normed linear spaces: a classification of the different concepts and some open problems, Rev. Mat., 2 (1989), 53-57.
[2] J. Anderson, On normal derivations, Proc. Amer. Math. Soc., 38 (1973), 135-140.
[3] F. Kittaneh, Normal derivations in normal ideals, Proc. Amer. Math. Soc., 6 (1995), 1979-1985.
[4] S. Mecheri, On the range and kernel of the elementary operators $\sum_{i=1}^{n}A_{i}XB_{i}-X$}, Acta Math. Univ. Comnianae, 52 (2003), 119-126.
[5] A. Bachir, A. Segres, Numerical range and orthogonality in normed spaces, Filomat, 23 (2009), 21-41.
[6] G. Bachman, L. Narici, Functional Analysis, Academic press, New York, 2000.
[7] D. K. Bhattacharya, A. K. Maity, Semilinear tensor product of $\Gamma$- Banach algebras, Ganita, 40(2) (1989), 75-80.
[8] F. Bonsall, J. Duncan, Complete Normed Algebra, Springer Verlag, New York, 1973.
[9] J. O. Bonyo, J. O. Agure, Norm of a derivation and hyponormal operators, Int. J. Math. Anal., 4(14) (2010), 687-693.
[10] S. Bouali, Y. Bouhafsi, On the range of the elementary operator $X\mapsto AXA-X$, Math. Proc. R. Ir. Acad., 108 (2008), 1-6.
[11] M. Cabrera, A. Rodriguez, Nondegenerately ultraprime Jordan Banach algebras, Proc. London Math. Soc., 69 (1994), 576-604.
[12] J. A. Canavati, S. V. Djordjevic, B. P. Duggal, On the range closure of an elementary operator, Bull. Korean Math. Soc., 43 (2006), 671-677.
[13] H. K. Du, G. X. Ji, Norm attainability of elementary operators and derivations, Northeast. Math. J., 3 (1994), 394-400.
[14] H. K. Du, Y. Q. Wang, G. B. Gao, Norms of elementary operators, Proc. Amer. Math. Soc., 4 (2008), 1337-1348.
[15] T. K. Dutta, H. K. Nath, R. C. Kalita, $\alpha$-derivations and their norms in projective tensor products of $\Gamma$-Banach algebras, J. London Math. Soc., 2(2) (1998), 359-368.
[16] M. B. Franka, Tensor products of C*-algebras, operator spaces and Hilbert C*-modules, Math. Comm., 4 (1999), 257-268.
[17] P. Gajendragadkar, Norm of derivations of Von-Neumann algebra, J. Trans. Amer. Math. Soc., 170 (1972), 165-170.
[18] R. Kadison, C. Lance, J. Ringrose, Derivations and automorphisms of operator algebra II, Math. J. Funct. Anal., 1 (1967), 204-221.
[19] D. J. Keckic, Orthogonality in $C_{1}$ and $C_{\infty}$ spaces and normal derivations, J. Operator Theory, 51 (2004), 89-104.
[20] D. J. Keckic, Orthogonality of the range and kernel of some elementary operators, Proc. Amer. Math. Soc., 11 (2008), 3369-3377.
[21] E. Kreyzig, Introductory Functional Analysis with Applications, John Wiley and Sons, New York, 1978.
[22] J. Kyle, Norms of derivations, J. London Math. Soc., 16 (1997), 297-312.
[23] B. Magajna, The norm of a symmetric elementary operator, Proc. Amer. Math., Soc. 132 (2004), 1747-1754.
[24] M. Mathieu, Properties of the product of two derivations of a C*- algebra, Canad. Math. Bull., 42 (1990), 115-120.
[25] M. Mathieu, More properties of the product of two derivations of a C*- algebra, Canad. Math. Bull., 42 (1990), 115-120.
[26] M. Mathieu, Elementary operators on Calkin algebras, Irish Math. Soc. Bull., 46 (2001), 33-42.
[27] J. G. Murphy, C*-algebras and Operator Theory, Academic Press Inc., Oval Road, London, 1990.
[28] N. B. Okelo, J. O. Agure, D. O. Ambogo, Norms of elementary operators and characterization of norm - attainable operators, Int. J. Math. Anal., 24 (2010), 687-693.
[29] N. B. Okelo, J. O. Agure, P. O. Oleche, Various notions of orthogonalty in normed spaces, Acta Math. Sci., 33B(5) (2013) 1387–1397.
[30] A. Turnsek, Orthogonality in $\mathcal{C}_{p}$ classes, Monatsh. Math., 132 (2001), 349 - 354.
[31] J. P. Williams, Finite operators, Proc. Amer. Math. Soc., 26 (1970), 129- 135.

Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Makaleler
Yazarlar	Bernard Okelo
Yayımlanma Tarihi	24 Aralık 2018
Gönderilme Tarihi	11 Temmuz 2018
Kabul Tarihi	16 Ağustos 2018
Yayımlandığı Sayı	Yıl 2018

Kaynak Göster

APA	Okelo, B. (2018). Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators. Communications in Advanced Mathematical Sciences, 1(2), 91-98. https://doi.org/10.33434/cams.442556
AMA	Okelo B. Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators. Communications in Advanced Mathematical Sciences. Aralık 2018;1(2):91-98. doi:10.33434/cams.442556
Chicago	Okelo, Bernard. “Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators”. Communications in Advanced Mathematical Sciences 1, sy. 2 (Aralık 2018): 91-98. https://doi.org/10.33434/cams.442556.
EndNote	Okelo B (01 Aralık 2018) Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators. Communications in Advanced Mathematical Sciences 1 2 91–98.
IEEE	B. Okelo, “Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators”, Communications in Advanced Mathematical Sciences, c. 1, sy. 2, ss. 91–98, 2018, doi: 10.33434/cams.442556.
ISNAD	Okelo, Bernard. “Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators”. Communications in Advanced Mathematical Sciences 1/2 (Aralık 2018), 91-98. https://doi.org/10.33434/cams.442556.
JAMA	Okelo B. Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators. Communications in Advanced Mathematical Sciences. 2018;1:91–98.
MLA	Okelo, Bernard. “Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators”. Communications in Advanced Mathematical Sciences, c. 1, sy. 2, 2018, ss. 91-98, doi:10.33434/cams.442556.
Vancouver	Okelo B. Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators. Communications in Advanced Mathematical Sciences. 2018;1(2):91-8.

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