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Abstract For an n-by-n matrix A, let w ( A ) and ‖ A ‖ denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening… Click to show full abstract
Abstract For an n-by-n matrix A, let w ( A ) and ‖ A ‖ denote its numerical radius and operator norm, respectively. The following three inequalities, each a strengthening of w ( A ) ≤ ‖ A ‖ , are known to hold: w ( A ) 2 ≤ ( ‖ A ‖ 2 + w ( A 2 ) ) / 2 , w ( A ) ≤ ( ‖ A ‖ + ‖ A 2 ‖ 1 / 2 ) / 2 , and w ( A ) ≤ ( ‖ A ‖ + w ( Δ t ( A ) ) ) / 2 ( 0 ≤ t ≤ 1 ), where Δ t ( A ) is the generalized Aluthge transform of A. In this paper, we derive necessary and sufficient conditions in terms of the operator structure of A for which the inequalities become equalities.
Journal Title: Linear Algebra and its Applications
Year Published: 2018
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