Gap Between Operator Norm and Spectral Radius for the Square of Antidiagonal Block Operator Matrices

Year 2022, Volume: 5 Issue: 1, 1 - 7, 17.03.2022

Elif Otkun Çevik

https://doi.org/10.33434/cams.1022686

Abstract

In this work, the gap between operator norm and spectral radius for the square of antidiagonal block operator matrices in the direct sum of Banach spaces has been investigated, and also the gap between operator norm and numerical radius for the square of same matrices in the direct sum of Hilbert spaces has been studied.

Keywords

Antidiagonal operator matrix, Numerical radius, Operator norm, Spectral radius

References

• [1] P. R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York, 1982.
• [2] K. E. Gustafson, D. K. M. Rao, Numerical Range: The field of values of linear operators and matrices, Springer-Verlag, New York, 1982.
• [3] F. Kittaneh, Spectral radius inequalities for Hilbert space operators, Proc. Am. Math. Soc., 134(2) (2006), 385-390.
• [4] A. Abu-Omar, F. Kittaneh, Estimates for the numerical radius and spectral radius of the Frobenius companion matrix and bounds for the zeros of polynomials, Ann. Func. Anal., 5(1) (2014), 56-62.
• [5] A. Abu-Omar, F. Kittaneh, Upper and lower bounds for the numerical radius with an application to involution operators, Rocky Mt. J. Math., 45(4) (2015), 1055-1065.
• [6] A. Abu-Omar, F. Kittaneh, Notes on some spectral radius and numerical radius inequalities, Studia Math., 227(2) (2015), 97-109.
• [7] H. Guelfen, F. Kittaneh, On numerical radius inequalities for operator matrices, Numer. Funct. Anal. Optim., 40(11) (2019), 1231-1241.
• [8] Y. Bedrani, F. Kittaneh, M. Sabbaheh, Numerical radii of accretive matrices, Linear Multilinear Algebra, 69(5) (2021), 957-970.
• [9] W. Bani-Domi, F. Kittaneh, Norm and numerical radius inequalities for Hilbert space operators, Linear Multilinear Algebra, 69(5) (2021), 934-945.
• [10] M. Zima, A theorem on the spectral-radius of the sum of 2 operators and its application, Bull. Aust. Math. Soc., 48(3) (1993), 427-434.
• [11] M. Zima, On the local spectral radius of positive operators, Proc. Am. Math. Soc., 131(3) (2003), 845-850.
• [12] M. Zima, On the local spectral radius in partially ordered Banach spaces, Czechoslov. Math. J., 49(4) (1999), 835-841.
• [13] M. Zima, Spectral radius inequalities for positive commutators, Czechoslov. Math. J., 64(1) (2014), 1-10.
• [14] M. Gurdal, M. T. Garayev, S. Saltan, Some concrete operators and their properties, Turk. J. Math., 39(6) (2015), 970-989.
• [15] M. T. Karaev, N. S. Iskenderov, Numerical range and numerical radius for some operators, Linear Algebra Appl., 432(12) (2010), 3149-3158.
• [16] M. Demuth, Mathematical aspect of physics with non-selfadjoint operators, List of open problem (American Institute of Mathematics Workshop, 8-12 June), 2015.
• [17] J. Lindenstrauss , L. Tzafriri, Classical Banach Spaces, First ed., Springer-Verlag, 1977.
• [18] M. A. Naimark, S. V. Fomin, Continuous direct sums of Hilbert spaces and some of their applications, Uspehi. Mat. Nauk., 10 (1955), 111-142.
• [19] E. O. Çevik, Z. I. Ismailov, Spectrum of the direct sum of operators, Electron. J. Differ. Equ., 210 (2012), 1-8.
• [20] E. O. Çevik, Some numerical characteristics of direct sum of operators, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 692 (2020), 1221-1227.