The Convergence of Some Spectral Characteristics on the Convergent Series

Öz

In this study, convergence properties of spectral, numerical and Crawford gap functions via convergences of Hilbert space operator series in difference and ratio cases are investigated. Obtained results have been applied to some classes continuous functions of the operators.

Anahtar Kelimeler

• Operator Norm
• Spectral Radius
• Numerical Radius
• Crawford Number

Yakınsak Seriler Üzerinde Bazı Spektral Karakteristiklerin Yakınsaması

Öz

Bu çalışmada, fark ve oran durumlarında yakınsak Hilbert uzay operatör serileri üzerinden spektral, sayısal ve Crawford boşluk fonksiyonlarının yakınsama özellikleri incelenmiştir. Elde edilen sonuçlar operatörlerin bazı sürekli fonksiyon sınıflarına uygulanmıştır.

Anahtar Kelimeler

• Operatör Normu
• Spektral Yarıçap
• Sayısal Yarıçap
• Crawford Sayısı

Kaynakça

1. Gelfand, M. (1941). Normierte ringe. Matematicheskii Sbornik, 9(51), 3-24.
2. Halmos, P. R. (1982). A Hilbert Space Problem Book. Springer-Verlag, New York.
3. Yamazaki, T. (2007). On upper and lower bounds of the numerical radius and equality condition. Studia Math., 178, 83-89.
4. Gustafson, K. E., & Rao, D. K. M. (1997). Numerical Range: The Field of Values of Linear Operators and Matrices. Springer, New York.
5. Bani-Domi, W., & Kittaneh, F. (2021). Refined and generalized numerical radius inequalities for 2x2 operator matrices. Linear Algebra Appl., 624, 364-386.
6. Bhunia, P., & Paul, K. (2021). New upper bounds for the numerical radius of Hilbert space operators. Bull. Sci. Math., 167, 1-11.
7. Bhunia, P., Paul K., & Nayak, R. K. (2021). Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices. Math. Inequal. Appl., 24, 167-183.
8. Kittaneh, F. (2003). A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math., 158, 11-17.

9. Kittaneh, F. (2005). Numerical radius inequalities for Hilbert space operators. Studia Math., 168, 73-80.
10. Du, K. (2010). The ratios between the spectral norm, the numerical radius and the spectral radius. Int. J. Comput. Math. Sci., 4(8), 388-391.
11. Al-Hawari, M. (2013). The ratios between the numerical radius and spectral radius of a matrix and the square root of the spectral norm of the square of this matrix. International Journal of Pure and Applied Mathematics, 82(1), 125-131.
12. Demuth, M. (2015). Mathematical aspect of physics with non-selfadjoint operators, list of open problem. American Institute of Mathematics Workshop. 8-12 June, Germany.
13. Ismailov, Z. I., & Ipek Al, P. (2021). Gaps between operator norm and spectral and numerical radii of the Tensor product of operators. Turkish Journal of Analysis and Number Theory, 9(2), 22-24.
14. Ismailov, Z. I., & Ipek Al, P. (2022). Some spectral characteristic numbers direct sum of operators. Turkish Journal of Analysis and Number Theory, 10(1), 21-26.
15. Ismailov, Z. I., & Otkun Çevik, E. (2022). On the one spectral relation for analytic function of operator. Journal of Nonlinear Science and Applications, 15, 301-307.
16. Ismailov, Z. I., & Mert, R. Ö. (2022). Gaps between some spectral characteristics of direct sum of Hilbert space operators. Operators and Matrices, 16(1), 337-347.
17. Otkun Çevik, E., & Ismailov, Z. I. (2021). Spectral radius and operator norm of nxn triangular block operator matrices. Journal of Analysis and Number Theory, 9(1), 7-12.
18. Otkun Çevik, E., & Ismailov, Z. I. (2021). Spectral radius and operator norm of infinite upper triangular double band block operator matrices. Journal of Analysis and Number Theory, 9(2), 19-22.
19. Bachman, G., & Narici, L. (1966). Functional Analysis. Academic Press, New York.
20. Gürdal, M., Garayev, M. T., & Saltan S. (2015). Some concrete operators and their properties. Turkish Journal of Mathematics, 39, 970-989.
21. Aleksandrov, A. B., & Peller, V. V. (2010). Operator Hölder–Zygmund functions. Advances in Mathematics, 224, 910-966.