Gaps between Operator Norm and Spectral and Numerical Radii of the Tensor Product of Operators

Zameddin I. Ismailov^1, and Pembe Ipek Al¹

¹Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey

Cite this paper:
Zameddin I. Ismailov and Pembe Ipek Al. Gaps between Operator Norm and Spectral and Numerical Radii of the Tensor Product of Operators. Turkish Journal of Analysis and Number Theory. 2021; 9(2):22-24. doi: 10.12691/tjant-9-2-1

Abstract

One of the fundamental problem of the Spectral Theory of Linear Operators is to determine of the geometric place of the spectrum of the given operator and calculate the spectral and numerical radii of this operator. Other important problem in this theory is to explained the situation the spectral (numerical) radius is equal or not to operator norm. The only known way to calculate the spectral radius to date is the classical Gelfand formula, which often presents great technical challenges. Also, there is not yet a method of calculating the numerical radius for an operator. It should be noted that the finding the numerical range and numerical radius means maximally localizing the spectrum of an operator. The main purpose of this paper is to determine the relations gaps between operator norm and spectral and numerical radii of the tensor product operators associated with the compatible gaps of coordinate operators.

Keywords:
operator norm spectral radius numerical radius

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	I. M. Gelfand, Normierte ringe, Matematicheskii Sbornik, 9, 51 (1941), 3-24.
[2]	F. Hirsh, G. Lacombe, Elements of functional analysis, Springer, New York, 1999.
[3]	K. E. Gustafson, D. K. M. Rao, Numerical Range: The field of values of linear operators and matrices, Springer, New York, 1997.
[4]	J. H. Anderson, C. Foias, Properties which normal operators share with normal derivations and related operators, Pacific Journal of Mathematics, 61, 2, 1975, 313-325.
[5]	P. R. Halmos, A Hilbert space problem book, Van Nostrand, New York, 1967.
[6]	M. H. Stone, Linear transformations in Hilbert Space and their applications to analysis, Amer. Math. Soc. Colloquium Publications, 5, New York, 1932.
[7]	M. Demuth, Mathematical aspect of physics with non-selfadjoint operators, List of open problem, American Institute of Mathematics Workshop, 8-12 June 2015.
[8]	Y. M. Berezanskii, Z. G. Sheftel, G. F. Us, Functional analysis II, Birkhauser Verlag, Basel Switzerland, 1990.
[9]	A. Brown, C. Pearchy, Spectra of tensor products of operator, Proceedings of the American Mathematical Society 17, 1, 1966, 162-166.
[10]	M. Gürdal, M. T. Garayev, S. Saltan, Some concrete operators and their properties, Turkish Journal of Mathematics, 39, 2015, 970-989.