Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Upper Bounds for the Euclidean Operator Radius and Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 472146 20 pages doi 2008 472146 Research Article Upper Bounds for the Euclidean Operator Radius and Applications S. S. Dragomir Research Group in Mathematical Inequalities Applications School of Engineering Science Victoria University . Box 14428 Melbourne VIC 8001 Australia Correspondence should be addressed to S. S. Dragomir Received 5 September 2008 Accepted 3 December 2008 Recommended by Andros Ronta The main aim of the present paper is to establish various sharp upper bounds for the Euclidean operator radius of an n-tuple of bounded linear operators on a Hilbert space. The tools used are provided by several generalizations of Bessel inequality due to Boas-Bellman Bombieri and the author. Natural applications for the norm and the numerical radius of bounded linear operators on Hilbert spaces are also given. Copyright 2008 S. S. Dragomir. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Following Popescu s work 1 we present here some basic properties of the Euclidean operator radius of an n-tuple of operators T1 . Tn that are defined on a Hilbert space H . This radius is defined by n A1 2 We T1 . Tn sup V I Tih h 2 l h 1 1 We can also consider the following norm and spectral radius on B H n B H X xB H by setting 1 11 71 . . . Tn e sup lnTn Ằ1 . Ằn G n re sup r Ã1T1 fnTn V1 . M eB 2 Journal of Inequalities and Applications where r T denotes the usual spectral radius of an operator T e B H and Bn is the closed unit ball in Cn. Notice that - e is a norm on B H nn rp T- rp rp IK T1 . Tn lie IK Tl . Tn lie r4T1 . T Te T 1 . . . Tn . Now if we denote by II T1 . Tn the square root of the norm Il2n 1 TiT that is II II n T T i 1 1 2 then we
