Norms of certain Jordan elementary operators |
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Authors: | Xiaoli Zhang |
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Institution: | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, PR China |
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Abstract: | Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A,B∈B(H), the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, ∀X∈B(H). In this short note, we discuss the norm of UA,B. We show that if dimH=2 and ‖UA,B‖=‖A‖‖B‖, then either AB∗ or B∗A is 0. We give some examples of Jordan elementary operators UA,B such that ‖UA,B‖=‖A‖‖B‖ but AB∗≠0 and B∗A≠0, which answer negatively a question posed by M. Boumazgour in M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393]. |
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Keywords: | Jordan elementary operator Norm Numerical range |
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