On Fuglede-Putnam property and orthogonality for derivations induced by hyponormal operators

Kagali, David; Okelo, Benard; Owino, Julia

doi:10.30495/maca.2025.2073114.1150

On Fuglede-Putnam property and orthogonality for derivations induced by hyponormal operators

https://doi.org/10.30495/maca.2025.2073114.1150

Volume 7, Issue 4
Autumn 2025

Pages 61-78

Mathematical Analysis and its Contemporary Applications

Document Type : Original Article

Authors

David Kagali

Benard Okelo

Julia Owino

Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Box 210-40601, Bondo, Kenya

Abstract

The orthogonality of derivations induced by operators is an area with various applications in light of ever-dynamic technological advances. There are different types of orthogonality, and interesting results have emerged in which operators satisfying given conditions are chosen to establish Range-Kernel orthogonality. However, most of the results have focused on one type of orthogonality called the Birkhoff orthogonality. We have also herein considered the Birkhoff concept of orthogonality. Researchers have repeatedly posed the following question: Could there be a possibility for studying other types of orthogonality with respect to the range and the kernel of derivations apart from the Birkhoff orthogonality? In this note, we establish orthogonality conditions for derivations when implemented by hyponormal operators underthe Fuglede-Putnam property.

Keywords

Elementary operator

Eigenvalue

Orthogonality

[1] B. Ahmed, S. Durairaj, and S. Nawal, A Fuglede-Putnam property for $N$ -class A(k) operators, Aims Math., 5(6) (2020), 7458–7466.

[2] N. Aissa, Versions of Fuglede-Putnam theorems on p,w-Hyponormal operators, J. Universal Math., 5(1) (2022), 4–50.

[3] N. Aissa, A new class of non-normal operators on Hilbert spaces, TWMS Appl. Engin. Math., 13(1) (2023), 276–284.

[4] L. Arambasic and R. Rajna, A strong version of Birkhoff-James orthogonality in Hilbert $C *$ -modules, Ann. Funct. Anal., 5 (2014), 109–120.

[5] A. Seddik, On numerical range of elementary operators, Linear Alg. Appl., 338 (2001), 239–244.

[6] T. Ando and C. Li, The numerical range and numerical radius, Linear Multilinear Alg., 37 (1994), 1–3.

[7] E. Andrew and W. Green, Spectral Theory of Operators on Hilbert Spaces, School of Mathematics, Georgia Institute of Technology, 2002.

[8] I. Ali, H. Yang, and A. Shakoor, Generalized norms inequalities for absolute value operators, Int. J. Anal. Appl., 5 (2014), 1–9.

[9] P. M. Bajracharya and B. P. Ojha, Birkhoff-orthogonality and different particular cases of Carlsson's orthogonality in normed linear spaces, J. Math. Stats., 44 (2020), 14–26.

[10] M. Barraa and J. R. Gilles, On the Numerical range of compact operators on a Hilbert space, J. Lond. Math. Soc., 55 (1972), 704–706.

[11] I. Bhatacharya and G. Priyanka, Characterization of Birkhoff-James orthogonality, J. Math. Anal. Appl., 407 (2013), 350–358.

[12] N. Bebiano, On generalized numerical range of operators on an indefinite inner product Space, Linear Multilinear Alg., 52 (2004), 203–233.

[13] M. Barraa, Essential numerical range of elementary operators, Proc. Amer. Math. Soc., 133 (2004), 1723–1726.

[14] C. Benitez, Orthogonality in normed linear spaces: A classification of the different concepts and some open problems, Rev. Mat. Univ. Complut. Madrid, 2 (1989), 53–57.

[15] D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal., 99 (1991), 262–292.

[16] P. Bourdon and J. Shapiro, What is the numerical range of composition operator, Manatsh. Math., 43 (2000), 65–79.

[17] D. Chandler, The Toeplitz-Hausdorff theorem explained, Canad. Math. Bull., 14 (1971), 17–32.

[18] J. Chmielinski, On an $ϵ$ -Birkhoff-James Orthogonality, J. Ineq. Pure. App. Math., 79 (2005), 28–42.

[19] L. Chi-Kwong, Numerical ranges of an operator in an indefinite inner product space, Electr. J. Linear Alg., 1 (1996), 1–17.

[20] G. Choi and S. K. Kim, The Birkhoff-James orthogonality and norm attainment for multilinear maps, J. Math. Anal. App., 502 (2021), 125–275.

[21] S. Christoph, On isolated points of the spectrum of bounded linear operator, Proc. Amer. Math. Soc., 117 (1993), 32–45.

[22] W. Donoghue, On the numerical range of a bounded operator, Mich. Math. J., 4 (1962), 261–263.

[23] D. Hilbert, Spectral theory, Mich. Math. J., 21(1902), 1–14.

[24] E. B. Davis, Parameter Semigroups, Acad. New York, 1980.

[25] M. Embry, The numerical range of an operator, Pacific J. Math., 32 (1970), 97–107.

[26] B. Enrico, The Drazin spectrum of tensor product of Banach algebra elements and elementary operators, Linear Multilinear Alg., 61 (2013), 295–307.

[27] L. A. Fialkow, Spectral properties of elementary operators, Acta Sci. Math., 46 (1983), 269–282.

[28] M. R. Flora and F. O. Nyamwala, On the Maximal numerical range of elementary operators, Int. J. Pure. Appl. Math., 112 (2017), 112–114.

[29] M. R. Flora, On the essential numerical range of elementary operators, Int. J. Pure. Appl. Math., 123 (2017), 741–747.

[30] M. Miguel, M. Javier, Q. Alicia, S. Roy, and D. Sain, A Numerical range approach to Birkhoff-James orthogonality with applications, Mat. Sub. Clas., 3 (2023), 1–7.

[31] F. O. Nyamwala and J. O. Agure, On norms of elementary operators in Banach algebras, Int. J. Math. Anal., 2 (2008), 411–424.

[32] H. Gau and P. Wu, Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl., 364 (2003), 213–222.

[33] P. R. Halmos, Numerical ranges and normal dilations, Acta Math. Szeged, 25 (1964), 1–8.

[34] P. R. Halmos, Hilbert Space Operators and Numerical Ranges, Van Nostrand, Princeton, 1967.

[35] P. R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York, 1982.

[36] R. E. Harte, Spectral mapping theorems, Proc. R. Ir. Acad., 72 (1972), 89–107.

[37] K. Hubert, The Numerical Range and Spectrum of a product of two orthogonal Projections, Hal. Arch. Fr., 23 (2013), 338–341.

[38] G. Hwa-Long and W. Pei, Excursions in numerical ranges, Bull. Inst. Math. Acad. Sinica, 9 (2014), 351–370.

[39] J. Jamison and F. Botelho, On elementary operators and the Aluthge transform, Linear Algebra Appl., 432 (2010), 275–285.

[40] D. Jiu and A. Zhou, A spectrum theorem for perturbed bounded linear operators, Appl. Math. Comput., 201 (2008), 723–728.

[41] C. R. Johnson, Computation of field of values of a 2 $\times$ 2 matrix, J. Res. NBS, 78 (1974), 105–108.

[42] C. R. Johnson, Normality and the numerical range, Linear Algebra Appl., 15 (1976), 89–94.

[43] C. R. Johnson, Gersgovin sets and the field of values, J. Math. Anal. and Appl., 45 (1974), 416–419.

[44] N. Johnston, Moein, R. Pereira, and S. Plosker, Birkhoff-James orthogonality in the trace norm with applications to quantum resource theories, Int. J. Math. Anal., 13 (2016), 121–135.

[45] E. Kreyzig, Introduction to Functional Analysis with Applications, John Wiley and Sons, 1978.

[46] L. A. Fialkow, Essential spectra of elementary operators, Trans. Amer. Math. Soc., 267 (1981), 123–135.

[47] L. Molnar and P. Semrl, Elementary operators on self-adjoint operators, J. Math. Anal. Appl., 327 (2007), 302–309.

[48] B. Magajna, Hilbert Modules and Tensor products of operator spaces, Banach Centree Publications, Warszawa, 1997.

[49] M. Martin, On different definitions of numerical range, J. Math. Anal., 87 (2015), 98–107.

[50] M. Mathieu, Elementary Operators on prime $C *$ -algebras, Math. Ann., 284 (1989), 223–244.

[51] C. Meng, A condition that normal operator have a closed numerical range, Proc. Amer. Math. Soc., 8 (1957), 85–88.

[52] M. Boumazgour, A note concerning the numerical range of a basic elementary operator, Ann. Funct. Anal., 7 (2016), 434–441.

[53] B. B. Morrel, A decomposition for some operators, Indiana Univ. Math. J., 23 (1973), 497–511.

[54] B. Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, 1970.

[55] N. B. Okelo and S. Kisengo, On the numerical range and spectrum of normal operators on Hilbert spaces, Sci. Technol. Sci. Technol., 2 (2012), 59–65.

[56] N. B. Okelo, J. O. Agure, and D. O. Ambogo, Norms of elementary operators and characterization of norm attainable operators, Int. J. Math. Anal., 4 (2010), 1197–1204.

[57] N. B. Okelo, J. A. Otieno, and O. Ongati, On the numerical ranges of convexoid operators, Int. J. Mod. Sci. Technol., 2 (2017), 56–60.

[58] G. Priyanka and T. Sushil, Birkhoff-James orthogonality and applications: A survey, Math. Sub. Cls., 7 (1991), 78–90.

[59] C. Putnam, On the spectra of semi-normal operators, Tran. Amer. Math. Soc., 119 (1965), 509–523.

[60] W. Qin and H. Jinchuan, Point spectrum preserving elementary operators on $B (H)$ , Amer. Math. Soc., 126 (1998), 2083–2088.

[61] R. M. Timoney, Norm of elementary operators, Irish Math. Soc. Bull., 46 (2001), 13–17.

[62] R. Raghavendran, On Toeplitz-Hausdorff theorem on numerical ranges, Proc. Amer. Math. Soc., 20 (1969), 284–285.

[63] F. B. Saidi, An extension of the portion of orthogonality to Banach spaces, J. Math. Anal. Appl., 267 (2002), 29–47.

[64] A. Seddik, The numerical range of elementary operators, Lin. Mult. Alg., 338 (2001), 45–57.

[65] A. Seddik, On the Numerical range and norm of elementary operators, Lin. Mult. Alg., 52 (2004), 17–23.

[66] L. Vasie, A Fuglede Putnam theorem modulo the Hilbert-Schmidt class for almost normal operators with finite modulus of Hilbert-Schmidt quasi-triangularity, Sci. J., 19(3) (2017), 23–45.

[67] P. Wojcik, Approximate orthogonality in normed spaces and its applications II, Linear Algebra Appl., 632 (2022), 258–267.

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