Mathematica Bohemica, first online, pp. 1-20


Numerical radius inequalities for Hilbert $C^*$-modules

Sadaf Fakri Moghaddam, Alireza Kamel Mirmostafaee

Received May 10, 2021.   Published online January 26, 2022.

Abstract:  We present a new method for studying the numerical radius of bounded operators on Hilbert $C^*$-modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert $C^*$-module spaces.
Keywords:  numerical radius; inner product space; $C^*$-algebra
Classification MSC:  47A12, 46C05, 47C10
DOI:  10.21136/MB.2022.0066-21

PDF available at:  Institute of Mathematics CAS

References:
[1] P. Bhunia, S. Bag, K. Paul: Numerical radius inequalities and its applications in estimation of zeros of polynomials. Linear Algebra Appl. 573 (2019), 166-177. DOI 10.1016/j.laa.2019.03.017 | MR 3933295 | Zbl 07060568
[2] S. S. Dragomir: Some refinements of Schwarz inequality. Proceedings of the Simpozionul de Matematici si Aplicatii, Timisoara, Romania (1985), 13-16.
[3] S. S. Dragomir: A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces. Banach J. Math. Anal. 1 (2007), 154-175. DOI 10.15352/bjma/1240336213 | MR 2366098 | Zbl 1136.47006
[4] S. S. Dragomir: Inequalities for the norm and numerical radius of composite operator in Hilbert spaces. Inequalities and Applications International Series of Numerical Mathematics 157. Birkhäuser, Basel (2009), 135-146. DOI 10.1007/978-3-7643-8773-0_13 | MR 2758975 | Zbl 1266.26036
[5] S. S. Dragomir: Power inequalities for the numerical radius of a product of two operators in Hilbert spaces. Sarajevo J. Math. 5 (2009), 269-278. MR 2567758 | Zbl 1225.47008
[6] M. Goldberg, E. Tadmor: On the numerical radius and its applications. Linear Algebra Appl. 42 (1982), 263-284. DOI 10.1016/0024-3795(82)90155-0 | MR 0656430 | Zbl 0479.47002
[7] A. A. Goldstein, J. V. Ryff, L. E. Clarke: Problems and solutions: Solutions of advanced problems 5473. Am. Math. Mon. 75 (1968), 309-310. DOI 10.2307/2314992 | MR 1534789
[8] K. E. Gustafson, D. K. M. Rao: Numerical Range: The Field of Values of Linear Operators and Matrices. Universitext. Springer, New York (1997). DOI 10.1007/978-1-4613-8498-4 | MR 1417493 | Zbl 0874.47003
[9] G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). MR 0944909 | Zbl 0634.26008
[10] M. S. Hosseini, M. E. Omidvar, B. Moosavi, H. R. Moradi: Some inequalities for the numerical radius for Hilbert $C^*$-modules space operators. Georgian Math. J. 28 (2021), 255-260. DOI 10.1515/gmj-2019-2053 | MR 4235824 | Zbl 07339609
[11] R. V. Kadison, J. R. Ringrose: Fundamentals of the Theory of Operator Algebras. Vol. 1. Elementary Theory. Pure and Applied Mathematics 100. Academic Press, New York (1983). MR 0719020 | Zbl 0518.46046
[12] I. Kaplansky: Modules over operator algebras. Am. J. Math. 75 (1953), 839-858. DOI 10.2307/2372552 | MR 0058137 | Zbl 0051.09101
[13] F. Kittaneh: Notes on some inequalities for Hilbert space operators. Publ. Res. Inst. Math. Sci. 24 (1988), 283-293. DOI 10.2977/prims/1195175202 | MR 0944864 | Zbl 0655.47009
[14] F. Kittaneh: Numerical radius inequalities for Hilbert space operators. Stud. Math. 168 (2005), 73-80. DOI 10.4064/sm168-1-5 | MR 2133388 | Zbl 1072.47004
[15] E. C. Lance: Hilbert $C^*$-Module: A Toolkit for Operator Algebraists. London Mathematical Society Lecture Note Series 210. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511526206 | MR 1325694 | Zbl 0822.46080
[16] C. A. McCarthy: $C_p$. Isr. J. Math. 5 (1967), 249-271. DOI 10.1007/BF02771613 | MR 0225140 | Zbl 0156.37902
[17] M. Mehrazin, M. Amyari, M. E. Omidvar: A new type of numerical radius of operators on Hilbert $C^*$-module. Rend. Circ. Mat. Palermo (2) 69 (2020), 29-37. DOI 10.1007/s12215-018-0385-3 | MR 4148774 | Zbl 07193605
[18] A. K. Mirmostafaee, O. P. Rahpeyma, M. E. Omidvar: Numerical radius ineqalities for finite sums of operators. Demonstr. Math. 47 (2014), 963-970. DOI 10.2478/dema-2014-0076 | MR 3290398 | Zbl 1304.47007
[19] B. Moosavi, M. S. Hosseini: Some inequalities for the numerical radius for operators in Hilbert $C^*$-modules space. J. Inequal. Spec. Funct. 10 (2019), 77-84. MR 4016178
[20] G. J. Murphy: $C^*$-Algebras and Operator Theory. Academic Press, Boston (1990). MR 1074574 | Zbl 0714.46041
[21] W. L. Paschke: Inner product modules over $B^*$-algebras. Trans. Am. Math. Soc. 182 (1973), 443-468. DOI 10.1090/S0002-9947-1973-0355613-0 | MR 0355613 | Zbl 0239.46062
[22] M. A. Rieffel: Induced representations of $C^*$-algebras. Adv. Math. 13 (1974), 176-257. DOI 10.1016/0001-8708(74)90068-1 | MR 0353003 | Zbl 0284.46040
[23] M. Sattari, M. S. Moslehian, T. Yamazaki: Some generalized numerical radius ineqalities for Hilbert space operators. Linear Algebra Appl. 470 (2015), 216-227. DOI 10.1016/j.laa.2014.08.003 | MR 3314313 | Zbl 1322.47010
[24] T. Yamazaki: On upper and lower bounds of the numerical radius and an equality condition. Stud. Math. 178 (2007), 83-89. DOI 10.4064/sm178-1-5 | MR 2282491 | Zbl 1114.47003

Affiliations:   Sadaf Fakri Moghaddam, Alireza Kamel Mirmostafaee (corresponding author), Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran, e-mail: sadaf.moghadam4@gmail.com, mirmostafaee@gmail.com


 
PDF available at: