Abstract
The main aim of the present paper is to establish various sharp upper bounds for the Euclidean operator radius of an -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.
Highlights
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; ·, ·
Notice that · e is a norm on B H n : T1, . . . , Tn e
I1 we can present the following result due to Popescu 1 concerning some sharp inequalities between the norms T1, . . . , Tn and T1, . . . , Tn e
Summary
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; ·, ·. The inequality 1.15 is better than the first inequality in 1.10 which follows from Popescu’s first inequality in 1.9 It provides, for the case that B, C are the selfadjoint operators in the Cartesian decomposition of A, exactly the lower bound obtained by Kittaneh in 1.7 for the numerical radius w A. Motivated by the useful applications of the Euclidean operator radius concept in multivariable operator theory outlined in 1 , we establish in this paper various new sharp upper bounds for the general case n ≥ 2. The case n 2, which is of special interest since it generates for the Cartesian decomposition of a bounded linear operator various interesting results for the norm and the usual numerical radius, is carefully analyzed
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