Upper Bounds for the Euclidean Operator Radius and Applications
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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, P.O. Box 14428, Melbourne, VIC 8001, Australia Correspondence should be addressed to S. S. Dragomir, [email protected] Received 5 September 2008; Accepted 3 December 2008 Recommended by Andros ´ Ront´a 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 q 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
w e T 1 , . . . , Tn
1/2 n 2 Ti h, h . : sup h1
1.1
i1
We can also consider the following norm and spectral radius on BHn : BH × · · · × BH, by setting 1
T1 , . . . , Tn : e re T1 , . . . , Tn
sup λ1 ,...,λn ∈Bn
sup λ1 ,...,λn ∈Bn
λ1 T1 · · · λn Tn ,
r λ1 T1 · · · λn Tn ,
1.2
2
Journal of Inequalities and Applications
where rT denotes the usual spectral radius of an operator T ∈ BH and Bn is the closed unit ball in Cn . Notice that ·e is a norm on BHn :
T1 , . . . , Tn T ∗ , . . . , Tn∗ , 1 e e
re T1 , . . . , Tn re T1∗ , . . . , Tn∗ .
1.3
Now, if we denote by T1 , . . . , Tn the square root of the norm ni1 Ti Ti∗ , that is,
1/2 n
∗
T1 , . . . , Tn :
Ti Ti ,
i1
1.4
then we can present the following result due to Popescu 1 concerning some sharp inequalities between the norms T1 , . . . , Tn and T1 , . . . , Tn e . Theorem 1.1 see 1. If T1 , . . . , Tn ∈ BHn , then
1 √ T1 , . . . , Tn ≤ T1 , . . . , Tn e ≤ T1 , . . . , Tn , n
1.5
√ where the constants 1/ n and 1 are best possible in 1.5. Following 1, we list here some of the basic properties of the Euclidean operator radius of an n-tuple of operators T1 , . . . , Tn ∈ BHn . i we T1 , . . . , Tn 0 if and only if T1 · · · Tn 0; ii we λT1 , . . . , λTn |λ|we T1 , . . . , Tn for any λ ∈ C; iii we T1 T1 , . . . , Tn Tn ≤ we T1 , . . . , Tn we T1 , . . . , Tn ; iv we U∗ T1 U, . . . , U∗ Tn U we T1 , . . . , Tn for any unitary operator U : K → H; v we X ∗ T1 X, . . . , X ∗ Tn X ≤ X2 we T1 , . . . , Tn for
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